Method and apparatus for controllably positioning a solar concentrator

ABSTRACT

A method, apparatus, control system and computer program product are provided for controllably positioning the solar concentrator. The method, apparatus, control system and computer program product determine the respective errors generated by more and different error sources than prior techniques, including error sources selected from the group consisting of a gravitational residue error, an elevation transfer function error and an error attributable to atmospheric refraction. Based upon the respective errors, the method, apparatus, control system and computer program product determine an elevation command and an azimuth command to compensate for the vertical error and the horizontal error between the centerline of the solar concentrator and the sun reference vector such that the solar concentrator can be more precisely positioned, thereby improving the efficiency with which the solar concentrator collects solar energy.

FIELD OF THE INVENTION

The present invention relates generally to methods and apparatus forcontrollably positioning a solar concentrator and, more particularly, tomethods and apparatus for controllably positioning a solar concentratorthat take into account at least one of a gravitational residue error,azimuth transfer function, and elevation transfer function error and anerror due to atmospheric refraction.

BACKGROUND OF THE INVENTION

There is currently a large domestic and international market for cleannon-polluting generated grid and remote electrical power, such as theelectrical power generated by solar energy generating systems. Thisdemand is anticipated only to grow. For example, over the next 12 years,the average growth of power consumption in California is estimated to be700 MW per year, while Arizona consumers are anticipated to demand anadditional 200 MW per year between the years 2000 and 2010. In addition,the state of Nevada has one of the fastest growing energy needs percapita in the United States with its electrical needs estimated to growin excess of 200 MW per year. In the solar belt states alone, theestimated growth is anticipated to be more than 1000 MW per year.Furthermore, an estimate by the World Bank of the internationalelectrical market in the solar belt countries is for growth of more than2000 MW year.

Of this growth in power consumption, at least a portion will be solarenergy. For example, the state of Arizona has decreed that 1% of allgenerated electrical power must be solar generated. This requirementcreates the need for 350 MW of grid electricity in this state from solarenergy alone. Other states in the solar belt, such as California,Nevada, New Mexico, etc., have or are expecting similar legislation.

A variety of solar-to-electrical energy conversion systems have beendeveloped with the most cost-effective systems being concentrating solarenergy systems that focus the energy of the sun to a relatively smallarea. One exemplary concentrating solar energy system is a Stirling dishdeveloped by McDonnell Douglas Corporation. A Stirling dish includes aplurality of reflective facets disposed side-by-side upon a supportframe to define a reflective surface. The reflective surface typicallyhas a parabolic or spherical shape. The parabolic reflective surface ofthe Stirling dish concentrates the incident solar energy upon a powerconversion unit that is located at the focal point of the reflectivesurface. In this regard, the power conversion unit is generally mountedupon the distal end of a support arm that extends forwardly of theStirling dish. The support frame that carries the plurality ofreflective facets and the support arm that carries the power conversionunit are mounted upon a pedestal which, in turn, is secured to afoundation within the earth. The pedestal permits the Stirling dish tomove in both an azimuth rotational plane and an elevation rotationalplane such that the Stirling dish can track the sun throughout the day.Thus, the Stirling dish also generally includes an azimuthal drive andan elevational drive, including an elevation activator, for providingthe desired movement in response to azimuth and elevation commandsissued by a controller or the like. Typically, these commands attempt todrive one Stirling dish to a position at which a centerline defined bythe reflective surface is aligned with the sun. Other types ofconcentrating solar energy systems exist, however, including heliostatsand other sun tracking solar concentrators.

There are two general types of tracking control systems for use withconcentrating solar energy systems, namely, open loop and closed loopcontrol systems. In a closed loop tracking control system, a sun sensoris aligned to the centerline defined by the reflective surface. As such,the sun sensor generates error signals between the centerline of thereflective surface and the line-of-sight to the sun, i.e., the sunreference vector. While closed loop tracking control systems can beeffective, closed loop tracking control systems are generally quiteexpensive due to the addition of a sun sensor and the attendant cabling,additional interface electronics and increased operational andmaintenance costs attributable to the additional hardware. Further,closed loop tracking control systems have difficulty maintaining trackduring periods of cloud cover. In this regard, if the reflective surfaceis not pointing at the sun when the sun comes out from behind theclouds, the concentrating solar energy system may be damaged. As such,open loop tracking commands must be calculated during the period of timein which the sun is behind the clouds. Additionally, further problemsarise in instances in which the face or lens of the sun sensor becomesdirty, such as from dust, sand or other airborne particles. In thisregard, a sun sensor relies upon the shading of the solar cells toobtain an error voltage. As such, a sun sensor having a dirty lens willunevenly illuminate the solar cells which, in turn, creates trackingerrors and also loss of track during low sun irradiance levels. Closedloop tracking control systems also suffer from an additional cost ofaligning the sun sensor to the centerline defined by the reflectivesurface and maintaining this alignment over time. Furthermore, the sunsensor oftentimes serves as a roosting place for birds which can causeadditional problems, by altering the alignment of the sun sensor orsoiling the lens of the sun sensor. As such, most concentrating solarenergy systems utilize an open loop tracking control system.

In an open loop tracking control system, the position of the sun iscalculated by a set of ephemeris equations. The reflective surface isthen commanded to point toward the position of the sun. As a result ofthe open loop nature of this system, there is no feedback from the solarconcentrator that the reflective surface is actually pointing at thesun. Unfortunately, the command coordinate system, i.e., the coordinatesystem in which the commands that direct the position of the solarconcentrator are formulated, and the concentrator coordinate system,i.e., the actual coordinate system defined by the physical constructionof the solar concentrator, are generally somewhat misaligned. As such,the centerline defined by the reflective surface may not be pointingdirectly at the sun even though commands have been issued that wouldhave caused the reflective surface to point at the sun if theconcentrator coordinate system and the command coordinate system wereidentical. Typical sources of error that will cause the misalignment ofthe command coordinate system and the concentrator coordinate system arepedestal/foundation tilt errors, elevation pivot point manufacturingtolerances, azimuth and elevation reference errors, gravity bending ofthe structure, atmospheric bending of the sun rays, reflectivity surfacemisalignment errors, elevation actuator offset errors and errorsinherent in the mathematical model utilized by the control system.

The deleterious effect of these errors can be reduced by increasing themanufacturing tolerances and the installation tolerances. However, theincrease in these tolerances will greatly increase the cost of aconcentrating solar energy system such that the resulting concentratingsolar energy system will no longer be economically competitive witheither non-concentrating solar systems or conventional power systems.The deleterious effect of these errors can also be reduced by modifyingthe pointing commands that position the reflective surface in an attemptto compensate for the misalignment errors.

In this regard, some concentrating solar energy systems automaticallyprovide a fixed bias of correction in the azimuth direction and a fixedbias of correction in the elevation direction. While the application ofa fixed amount of error correction is somewhat helpful, the azimuth andelevation errors vary throughout the day and year as the position of thesun and the solar concentrator changes. In this regard, FIGS. 1A and 1Bdepict the elevation and azimuth track errors in degrees as a functionof azimuth position in degrees from due south. In this regard, both theazimuth and elevation track errors vary by about 0.6 degrees such thatincorporation of a fixed correction value, such as an average errorcorrection value, would still subject the concentrating solar energysystem to a substantial amount of uncorrected error in both the azimuthand elevation rotational planes.

In order to take into account variations in the azimuth and elevationerrors throughout the day, one system manually determines the errorsattributable to azimuth tilt and gravity bending. To determine theerrors attributable to azimuth tilt, a leveling device, such as aninclinometer, is placed on the upper part of the rotating structure,i.e., the structure that rotates about the pedestal, and the angle isnoted. The solar concentrator is then rotated between five and tendegrees and the tilt angle is again measured. This process is repeatedin relatively small steps until the solar concentrator is rotatedthrough 360 degrees. The data is then plotted and the tilt parametersare estimated from the plotted data. Similarly, the errors attributableto gravity bending are determined by mounting an inclinometer on theelevation rotating part of the solar concentrator, i.e., the structureabove the elevation pivot point. The solar concentrator is thencommanded to an angle and the actual angle is measured with theinclinometer. The solar concentrator is then commanded to anotherelevation angle which is between about five and ten degrees greater thanthe initial elevation angle. This process is continued until measurementdata has been obtained for elevation angles from 0 degrees to 90degrees. The measured angles are then subtracted from the commandedelevation angles to obtain the error attributable to gravity bending.This data is then curve fit as a function of the elevation angle and theresulting curve is utilized to adjust the commanded elevation angle inorder to correct for errors attributable to gravity bending.

While this technique takes into account the variations in the errorsattributable to azimuth tilt and gravity bending throughout the day,this technique does not recognize that these errors, as well as theother errors to which the solar concentrator is subjected, vary not onlythroughout the day, but also from day to day and season to season as therelative position of the sun in the sky changes. Additionally, thistechnique only takes into account two tracking errors, namely, errorsattributable to azimuth tilt and errors attributable to gravity bending,thereby ignoring the effects of a number of other error sources, such asazimuth reference error, errors attributable to reflective surfacenon-orthogonality, elevation reference position error, elevationrotational tilt errors and the like. Moreover, this technique requiressubstantial manual labor in the field in order to collect the necessarydata.

Another technique for measuring the misalignment errors and formodifying the pointing commands is described by U.S. Pat. No. 4,564,275to Kenneth W. Stone, the contents of which are incorporated herein intheir entirety. This technique automatically aligns one or moreheliostats by comparing the actual sun beam centroid position to acommand reference position to determine the error in the sun beamcentroid location. This is done several times over the day dependingupon the required accuracy. The sun beam centroid position error is thenanalyzed to correlate the error to errors in the track alignment systemof the heliostat. In this regard, the technique described by U.S. Pat.No. 4,564,275 takes into account errors attributable to thenon-orthogonality of the facets of the reflective surface, errors in theelevation and azimuth reference positions and errors in the azimuthrotational tilt. Based upon the errors, the command reference positionis updated to automatically correct for the errors. While the techniquedescribed by U.S. Pat. No. 4,564,275 represents a substantialimprovement in open loop tracking control systems for solarconcentrators, it would be desirable to further improve the open looptracking control system in order to even more accurately align thecenterline defined by a reflective surface with the sun referencevector, thereby capturing a greater percentage of the energy deliveredby the sun and increasing the efficiency of the concentrating solarenergy system.

SUMMARY OF THE INVENTION

A method, apparatus, control system and computer program product areprovided for controllably positioning the solar concentrator.Advantageously, the various embodiments of the present inventiondetermine the respective errors generated by more and different errorsources than prior techniques, including error sources selected from thegroup consisting of a gravitational residue error, an elevation transferfunction error and an error attributable to atmospheric refraction.Based upon the respective errors, the various embodiments of the presentinvention determine an elevation command and an azimuth command tocompensate for the vertical error and the horizontal error between thecenterline of the solar concentrator and the sun reference vector suchthat the solar concentrator can be more precisely positioned, therebyimproving the efficiency with which the solar concentrator collectssolar energy.

According to one embodiment of the present invention, a control systemis provided for positioning a solar concentrator. The control systemincludes an input section for receiving signals representative of thevertical error and the horizontal error between the centerline of thesolar concentrator and the sun reference vector. In one embodiment, theinput section repeatedly receives signals representative of the verticalerror and the horizontal error at a plurality of different timesthroughout the day such that the solar concentrator can be repositionedthroughout the day as necessary to optimize its collection of the solarenergy. The control system also includes a processing element fordetermining the elevation command and the azimuth command to compensatefor the vertical error and the horizontal error. In one advantageousembodiment, the processing element is comprised of a computer programproduct having a computer-readable storage medium with computer-readableprogram code embodied therein for performing the various functions ofthe processing element. However, the processing element can be embodiedin other manners, if so desired.

Regardless of its physical embodiment, the processing element determinesrespective errors generated by a plurality of error sources thatcontribute to the vertical error and/or the elevation error. Accordingto the present invention, the plurality of error sources include atleast one error source selected from a group consisting of agravitational residue error, an elevation transfer function error and anerror due to atmospheric refraction. In one embodiment, the processingelement determines the respective errors by collectively determining agravitational residue error g, an elevation transfer function error eand an error due to atmospheric refraction. The processing element canalso determine additional errors by individually determining each of afirst azimuth rotational tilt error γ₁, a second azimuth rotational tilterror γ2, a first elevation rotational tilt error φ, a second elevationrotational tilt error δ₁ and a reflective surface non-orthogonalityerror δ₂.

The processing element also determines the elevation command and theazimuth command at least partially based upon the respective errorsgenerated by the plurality of error sources. In one advantageousembodiment, the processing element determines the elevation command bydetermining an elevation command angle Ψc as follows: Ψ_(c) sin⁻¹(sin γ₁cos Ψr cos Φr−sin γ₂ cos Ψr sin Φr+sin Ψr )−dΨ wherein γ₁ and γ₂ areazimuth rotational tilt errors, Φr and Ψr are elevation and azimuthangles, respectively, in an inertial reference system that are requiredto align the centerline of the solar concentrator and a sun referencevector, and dΨ is a combination of the elevation transfer functionerror, the gravitational residual error, and the error due toatmospheric refraction.

Likewise, the processing element of this advantageous embodimentpreferably determines the azimuth command by determining an azimuthcommand angle Φc as follows:$\Phi_{c} = {{\cos^{- 1}\left\lbrack \frac{{A\quad C} + {BD}}{A^{2} + B^{2}} \right\rbrack} - \Phi_{e}}$

wherein A, B, C and D are defined as follows:

A=cos λ cos δ₂

B=cos δ₁ sin δ₂−sin δ₁ cos δ₂ sin λ

C=cos γ₁ cos Ψr cos Φr−sin γ₁ sin Ψr

D=sin γ₁ sin γ₂ cos Ψr cos Φr+cos γ₂ cos Ψr sin Φr+sin Ψr cos γ₁ sin γ₂

wherein λ is defined as follows:

λ=Ψ_(c)−Ψ_(e) +g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c))

wherein Ψc is an elevation command angle, Ψe is an elevation referenceposition error, g(Ψ) is a gravitational residue error, e(Ψ) is anelevation transfer function error and r(Ψ) is an error due toatmospheric refraction, and wherein δ₁ is an elevation rotational tilterror, δ₂ is a reflective surface non-orthogonality error, γ₁ and γ₂ areazimuth rotational tilt errors and Φr and Ψr are elevation and azimuthangles, respectively, in an inertial reference system that are requiredto align the centerline of the solar concentrator and a sun referencevector.

The control system also includes an output section for providing signalsrepresentative of the elevation command and the azimuth command. Theelevation command and the azimuth command can then be utilized tocontrollably position the solar concentrator to compensate for thevertical error and the horizontal error between the centerline of thesolar concentrator and sun reference vector.

According to another aspect of the present invention, an apparatus forcontrollably positioning the solar concentrator is provided thatincludes a measurement system for determining the vertical error and thehorizontal error between the centerline of the solar concentrator andthe sun reference vector, a processing element as described above inconjunction with the control system, and a positioning mechanism forpositioning the solar concentrator based upon the elevation command andthe azimuth command in order to compensate for the vertical error andthe horizontal error.

The measurement system may be either a sun sensor or a digital imageradiometer. Alternatively, the measurement system may be adapted to movethe solar concentrator from a nominal position to an aligned position atwhich the difference between the gas temperatures of each quadrant ofthe solar concentrator is minimized. Based upon the horizontal andvertical distances that the solar concentrator is moved from the nominalposition to the aligned position, the measurement system of thisembodiment can determine the horizontal and vertical errors,respectively. In yet another embodiment, the measurement system may beadapted to move the solar concentrator from a nominal position to analigned position at which the maximum power factor is obtained.According to this embodiment, the measuring system is further adapted todetermine the horizontal and vertical distances that the solarconcentrator is moved from the nominal position to the aligned positionin order to determine the horizontal and vertical errors, respectively.

According to another aspect of the present invention, a method ofcontrollably positioning the solar concentrator is provided. In thisregard, the method initially determines the vertical and horizontalerrors between the centerline of the solar concentrator and the sunreference vector. The method then determines an elevation command and anazimuth command to compensate for the vertical and horizontal errors,respectively. In this regard, the method initially determines respectiveerrors generated by a plurality of error sources that contribute to oneor more of the vertical and horizontal errors. As described above, theplurality of error sources include at least one of the gravitationalresidue error g, the elevation transfer function error e and the error rdue to atmospheric refraction. The elevation command and the azimuthcommand are then determined based at least in part upon the respectiveerrors generated by the plurality of error sources. Based upon theelevation command and the azimuth command, the solar concentrator can bepositioned to compensate for the vertical and horizontal errors.

By taking into account the errors generated by a number of error sourcesincluding error sources not previously considered by open loop trackingcontrol systems and methods, such as the gravitational residue error g,the elevation transfer function error e and the error r due toatmospheric refraction, the method, apparatus, control system andcomputer program product of the present invention can more preciselyposition the solar concentrator such that the centerline of the solarconcentrator is aligned with the sun reference vector. As such, theefficiency with which the solar concentrator collects the solar energyis improved. By repeating this process at a number of different timesthroughout the day, various embodiments of the present invention canrepeatedly optimize the performance of the solar concentrator throughoutthe day to obtain much better performance than conventional systems thatutilize the same average correction factor throughout the entire dayeveryday. Still further, by implementing much of the functionality ofthe open loop control methodology of the present invention in software,the resulting apparatus is competitive in cost with conventional systemsand can be readily reconfigured, if necessary.

BRIEF DESCRIPTION OF THE DRAWINGS

Having thus described the invention in general terms, reference will nowbe made to the accompanying drawings, which are not necessarily drawn toscale, and wherein:

FIGS. 1A and 1B are graphical representations of the elevation andazimuth track errors in degrees as a function of azimuth position indegrees from due south for a conventional solar concentrator;

FIG. 2 is a schematic perspective view of a solar concentrator accordingto one embodiment of the present invention;

FIG. 3 is a block diagram of a solar concentrator according to oneembodiment of the present invention using a sun sensor;

FIG. 4 is an operational block diagram depicting the steps taken toobtain measurement data according to one embodiment of the presentinvention;

FIG. 5 is an operational block diagram of the steps undertaken todetermine the track alignment parameters for a solar concentrator from avariety of error sources according to one embodiment of the presentinvention;

FIG. 6 is a pictorial representation of the relationship between theinertial reference system and the reference system of the solarconcentrator;

FIG. 7 is a functional representation of the coordinate transformationbetween the reference and inertial coordinate systems to the solarconcentrator;

FIG. 8 is the total transformation matrix from reference coordinates tothe actual coordinates of the solar concentrator;

FIG. 9 is a operational block diagram of the steps undertaken to adjustthe elevation and azimuth commands in order to compensate for thecontributions of the variety of error sources according to oneembodiment of the present invention;

FIG. 10 is a graphical representation of the elevation and azimuthangles of the sun throughout the day at summer solstice;

FIG. 11 is a graphical representation of the gravitational residue errorg, the elevation transfer function error e and the error due toatmospheric refraction r as a function of elevation angle;

FIG. 12 is a graphical representation of the combined true elevationerror dΨ_(tj) as a function of elevation angle;

FIG. 13 is a graphical representation of the true elevation errordΨ_(true) as a function of azimuth angle;

FIG. 14 is a graphical representation of the true and measured values ofthe vertical measurement error as a function of azimuth angle;

FIG. 15 is a graphical representation of the true and measured values ofthe horizontal measurement error as a function of azimuth angle;

FIGS. 16a and 16 b are graphical representations of the true andmeasured values of the elevation error as a function of azimuth angleand elevation angle, respectively;

FIG. 17 is a graphical representation depicting the estimated elevationerror as a function of elevation angle;

FIGS. 18a and 18 b are graphical representations of the horizontalmeasurement errors as a function of azimuth angle and elevation angle,respectively;

FIGS. 19a and 19 b are graphical representations of the curve fitrepresentative of the horizontal measurement error as a function ofazimuth angle and elevation angle, respectively;

FIGS. 20a and 20 b are graphical representations of the true andestimated vertical and horizontal errors, respectively, as a function ofazimuth angle;

FIGS. 21a and 21 b are graphical representations of the corrected anduncorrected vertical and horizontal errors, respectively, as a functionof azimuth angle; and

FIGS. 22a and 22 b are graphical representations of the corrected anduncorrected estimated vertical and horizontal errors, respectively, as afunction of azimuth angle.

DETAILED DESCRIPTION OF THE INVENTION

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which preferred embodimentsof the invention are shown. This invention may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein; rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the scope of the invention to those skilled in the art. Likenumbers refer to like elements throughout.

Referring now to FIG. 2, a solar concentrator 10 is depicted. As knownto those skilled in the art, a solar concentrator includes a reflectivesurface 12, typically comprised of a plurality of reflective facets,mounted upon a support frame 14. The facets are typically mounted uponthe support frame to define a generally parabolic or spherical surface.As such, solar energy incident upon the reflective surface will beconcentrated to a focal point disposed upon a centerline 15 defined bythe reflective surface. The solar concentrator therefore also includes asolar receiver 16, such as a Stirling engine, mounted at the focal pointof the reflective surface for receiving the concentrated solar energyand for converting the solar energy into mechanical energy such as bymeans of the alternating compression and expansion of a confined gas.Although not illustrated, the Stirling engine is typically connected toan electrical power generator to convert the mechanical output from theStirling engine into electrical energy.

The solar receiver 16 is mounted upon a support arm 18 which, in turn,is typically connected to the support frame 14 of the reflective surface12. The solar concentrator 10 also includes a pedestal 20 mounted upon afoundation for supporting the support frame and the support arm as wellas the reflective surface and the solar receiver carried thereby. Thepedestal is adapted to permit movement of the solar concentrator in anazimuth rotational plane, typically from a generally east-facingposition at sunrise to a generally west-facing position at sunset, andin an elevation rotational plane, typically from a horizon-facingposition at sunrise to a maximum solar elevation-facing position in themiddle of the day and back to a horizon-facing position at sunset, so asto track the sun. As such, the solar concentrator includes an azimuthaldrive mechanism for providing the movement in the azimuth rotationalplane and an elevational drive mechanism for providing the movement inthe elevational rotational plane. In this regard, the elevational drivemechanism includes a linear actuator 22 for providing controlledmovement in the elevational rotational plane. Further details of a solarconcentrator are provided by U.S. Pat. No. 4,583,520 to John J.Dietrich, et al. and U.S. Pat. No. 5,758,938 to Carl E. Osterwisch, thecontents of both of which are incorporated herein by reference in theirentirety. While one embodiment of a solar concentrator is depicted inFIG. 2 and described above, the solar concentrator can be configured inmany different manners, such as a heliostat or any other type ofconcentrating solar energy system known to those skilled in the art.

According to the present invention, an apparatus, method, control systemand computer program product are provided for controllably positioningthe solar concentrator 10 to reduce, if not eliminate, any offsetbetween the centerline 15 defined by the reflective surface 12 of thesolar concentrator and the sun reference vector, that is, the vectorrepresenting the direction in which the sun rays are traveling uponincidence with the solar concentrator. As illustrated in FIG. 3, theapparatus 24 of the present invention includes three primary components,namely, a measurement system 26, a control system 27 including aprocessing element 28 and a positioning mechanism 30.

As described in more detail below, the measurement system 26 determinesthe vertical error and the horizontal error between the centerline 15defined by the reflective surface 12 of the solar concentrator 10 andthe sun reference vector. In other words, the measurement systemdetermines the misalignment of the solar concentrator which is due tothe collective contribution of a plurality of different types of errors.The processing element 28 determines respective errors generated by eachof the different error sources that contribute to the vertical errorand/or the elevation error. According to the present invention, theplurality of error sources include at least one of a gravitation residueerror g(Ψ), an elevation transfer function error e(Ψ), and an error r(Ψ)due to atmospheric refraction. The processing element then determinesthe elevation command and the azimuth command that will compensate forthe vertical error and the horizontal error based at least in part uponthe respective errors generated by the plurality of error sources.Finally, the positioning mechanism 30 positions the solar concentratorbased upon the elevation command and the azimuth command in order tocompensate for the vertical error and the horizontal error. As such, thecenterline defined by the reflective surface of the solar concentratorwill be in alignment with the sun reference vector, thereby improvingthe efficiency with which the solar concentrator collects solar energy.

While the solar concentrator 10 can be positioned by a variety ofdifferent mechanisms without departing from the spirit and scope of thepresent invention, the positioning mechanism 30 typically includes amotor 32 and the azimuthal and elevational drive mechanisms 34, 36 inorder to appropriately position the solar concentrator in both theazimuth rotational plane and the elevational rotational plane. In thisregard, the elevational drive mechanism includes, among othercomponents, the linear actuator 22 that is responsive to the motor.

Various types of measurement systems 26 can be utilized to determine thevertical and horizontal errors between the centerline 15 defined by thereflective surface 12 of the solar concentrator 10 and the sun referencevector, i.e., the line of sight to the sun. In one embodiment, themeasurement system comprises a sun sensor mounted upon the solarconcentrator and aligned along the centerline defined by the reflectivesurface in order to determine the vertical error and the horizontalerror between the centerline defined by the reflective surface and thesun reference vector.

In this regard, FIG. 4 depicts the operational steps that are performedin order to determine the vertical and horizontal errors, collectivelytermed the tracking error, of the sun sensor. In this regard, the sunelevation and azimuth position is determined by a plurality of theephemeris equations based upon the time and date and the latitude andlongitude of the solar concentrator 10. Based upon the calculatedposition of the sun, the solar concentrator is commanded to a positionin which the centerline 15 defined by the reflective surface 12 ispointed toward the current position of the sun as known to those skilledin the art. Although the motor 32 and the associated azimuthal andelevational drive mechanisms 34, 36 drive the solar concentrator to thedesignated position, various error sources cause the solar concentratorto actually be misaligned somewhat relative to the sun reference vector.The sun sensor detects this misalignment and determines the trackingerror which includes both the vertical error and the horizontal errorbetween the centerline defined by the reflective surface and the sunreference vector. The foregoing process is typically repeated at aplurality of times throughout the day, such as at periodic intervals of15 to 30 minutes in one embodiment.

As an alternative to a sun sensor, the measurement system 26 can includea digital image radiometer as described by U.S. Pat. No. 5,493,392 toJames V. Blackmon, et al., the contents of which are also incorporatedherein in their entirety. Like the sun sensor, the digital imageradiometer is mounted upon the solar concentrator 10 and determines thetracking error, comprised of the vertical and horizontal errors, betweenthe centerline 15 defined by the reflective surface 12 of the solarconcentrator and the sun reference vector.

In those embodiments in which the solar concentrator 10 is a Stirlingdish, the solar receiver 16 is divided into four quadrants. If the solarflux is not evenly distributed over the solar receiver, each quadrantwill have a different gas temperature. As such, the measurement system26 can include a controller associated with the Stirling dish forcommanding the Stirling dish to point toward the calculated position ofthe sun as described above. The gas temperature is then measured foreach quadrant and the maximum temperature difference between the gastemperatures of the four quadrants is then determined. The controllerthen commands the Stirling dish to move slightly up and then down in theelevation rotational plane and east and west in the azimuth rotationalplane with the temperature difference between the gas temperatures ofeach quadrant determined in each position of the Stirling dish. Thisperturbation of the position of the Stirling dish is repeated until thelowest gas temperature difference is determined. The position of theStirling dish at which the gas temperature difference is the lowest isdetermined to be the optimal position at which the centerline 15 of theSterling dish is actually pointed at the sun. As such, the vertical andhorizontal errors are determined by the controller to be the verticaland horizontal differences between the optimal position of the Stirlingdish at which the Stirling dish is actually pointed at the sun and theoriginal position to which the Stirling dish was initially commanded.

Similarly, the measurement system 26 can be a controller that determinesthe power level of the solar concentrator 10. In this regard, the outputpower and, more particularly, the power level, of the solar concentratoris a good measure of the relative alignment of the solar concentratorwith the sun reference vector. The power level is defined by the outputpower provided by the solar concentrator divided by the sun insulation.As described above, the solar concentrator is commanded to point at thecalculated position of the sun based upon the current time and date. Theposition to which the solar concentrator is pointed is then perturbedalternately in the azimuth rotational plane and the elevation rotationalplane. The power level of the output power provided by the solarconcentrator is determined by the controller in each position. Theposition at which the solar concentrator provides the maximum powerlevel is determined to be the position at which the solar concentratoris actually aligned with the sun reference vector. As such, thecontroller determines the tracking error, i.e., the vertical andhorizontal errors, between the original position to which the solarconcentrator was commanded and the aligned position having the maximumpower level. In each of these instances, the measurement systempreferably repeats the process of determining the horizontal andvertical errors between the nominal position to which the solarconcentrator is initially commanded and the aligned position at whichthe centerline 15 defined by the reflective surface 12 of the solarconcentrator is actually pointed at the sun reference vector.

Thereafter, the control system 27 of the present invention whichincludes the processing element 28 utilizes the measurement data, i.e.,the vertical and horizontal errors, to estimate the error parameters asillustrated schematically in FIG. 5. In an open loop tracking controlsystem such as the control system of the present invention, there aremany error sources that can introduce errors and adversely affect theactual position that the solar concentrator assumes in response to theelevation and azimuth position commands. By determining the contributionof each of these error sources, the control system of the presentinvention permits these errors to be corrected as shown in FIG. 9 suchthat the solar concentrator is pointed directly at the sun and theefficiency with which solar energy is collected is increased.Advantageously, the method, apparatus, control system and computerprogram product of the present invention permit more and different to bedetermined and corrected than conventional open loop tracking controlsystems.

The error sources that are corrected by the various embodiments of thepresent invention include the gravitational residue error g(Ψ). Asdescribed above, the solar concentrator 10 consists of two majorcomponents, namely, the mirror reflectivity assembly and the powerconversion unit (PCU). These two components are heavy and are located atthe opposite end of the structure in order to balance the loads on theelevation and azimuth drives. Because of the geometry, the gravitationalweight is not balanced, therefore as the elevation angle changes, thegravitational moment about the pivot point changes. As the gravitationalmoment changes, the structural bending introduced by the gravitationalmoment changes. Although a nominal value of the gravity bending is addedto the gimbal position command to provide some compensation for thestructural bending, there generally remains a gravitational residueerror occasioned by structural bending that has gone uncompensatedbecause of the difference in the structural strength from one system tothe next.

Another error source is the elevation transfer function error e(Ψ). Theelevation actuator is a linear actuator. Because of the geometry withrespect to the elevation pivot point, however, the transfer functionfrom elevation angle to actuator movement is a nonlinear function. Thenonlinear transfer function is referenced to a particular geometricposition of the actuator. During installation the actuator is set at thereference position. There is some error in finding this position,however, which results in an error between the commanded elevation angleand the actual elevation angle.

Another error source relates to atmospheric refraction r(Ψ). The sun'srays bend toward the earth's surface as they enter the atmosphere. Theamount of bending depends upon the elevation angle of the sun, theatmospheric pressure along the path of the sun's rays, and the airtemperature along the path of the sun. There are atmospheric refractionmodels to correct the bending of the sun's rays and these can beincorporated into the commanded elevation angle. However, there remainssome error residual that serves as this error source.

Azimuth rotational tilt γ₁ and γ₂ is another error source. There are anumber of error sources that can make the azimuth rotational plane notperpendicular to the local vertical. For example, not installing thepedestal exactly vertical is the most common source of this error, butother causes could be that the azimuth drive interface with the pedestalis at an angle due to machining tolerances, the foundation could settleafter installation, the azimuth drive bearing could be at an angle tothe vertical axis due to machining tolerance, etc.

Another error source is the elevation rotational tilt Φe and δ₁. Thereare also a number of error sources that can result in the elevationrotational plane not being the same as the local vertical plane. Theazimuth position of the concentrator must be referenced to the localSouth position. If it is not, the elevation rotational plane at a zeroelevation angle is not aligned with the command plane. Anothercontributor to this error is if the elevation pivot point is not alignedto the azimuth rotational plane due to structural/machining tolerance.Therefore the vertical axis of the elevation plane is not aligned to thevertical axis of the local vertical plane.

Reflective surface non-orthogonality δ₂ is another error source. Thecenterline defined by the reflective surface may not be parallel to thecenterline of the drive system. This type of error could result from thereflective surface support structural tolerance or misalignment of themirrors. The reflective surface is made up of 82 or more mirrorsegments. Each mirror facet must be aligned at specific cant angles. TheDigital Image Radiometer described by U.S. Pat. No. 4,564,275 can beused to align the mirror facets. If this system is not aligned exactlyto the concentrator drive system coordinate system, a non-orthogonalityerror will be introduced. Also the Digital Image Radiometer can onlyalign the mirror centerline to a specified accuracy. Since each mirrorhas some randomness associated with it, the effective centerline of thereflective surface will be misaligned or non-orthogonal to the drivecoordinate system.

One additional error source is due to the elevation reference positionerrors Ψ_(e) The elevation angle of the sun is calculated with respectto the local horizontal plane. If the actual elevation angle of thesolar concentrator is not determined with respect to this localhorizontal plane, an elevation reference track error occurs.

There are several coordinate systems used in the tracking system asillustrated in FIG. 6. The sun position is calculated in an inertialreference system (X_(r), Y_(r), Z_(r)) where the x-axis is pointingSouth, the y-axis is pointing East, and the z-axis is the localvertical. The second coordinate system is the tracker system (X_(t),Y_(t), Z_(t)) where X_(t) is aligned with the normal of the reflectivesurface, Z_(t) is in the vertical plane of the solar concentrator, andY_(t) and Z_(t) are in the plane of the aperture surface of the solarconcentrator.

The sun position is defined in an inertial reference system (r) by thetwo angles Φr and Ψr. The direction cosines of the sun reference vectorare: $\begin{matrix}{\begin{bmatrix}x_{r} \\y_{r} \\z_{r}\end{bmatrix} = \begin{bmatrix}{\cos \quad \psi_{r}\cos \quad \varphi_{r}} \\{\cos \quad \psi_{r}\sin \quad \varphi_{r}} \\{\sin \quad \psi_{r}}\end{bmatrix}} & (1)\end{matrix}$

where

Φ_(r)=required azimuth angle to point at the sun

Ψ_(r)=required elevation angle to point at the sun

The function of the control system is to point the centerline of thereflective surface (e.g., the X_(t) axis) at the sun. This isaccomplished by calculating azimuth and elevation command angles thatwill place the centerline of the reflective surface pointing at the sunas shown in FIG. 6. The centerline of the reflective surface or commandvector would then be co-linear with the sun vector, Equation (1). Thus,the command vector would be: $\begin{matrix}{\begin{bmatrix}x_{c} \\y_{c} \\z_{c}\end{bmatrix} = \begin{bmatrix}{\cos \quad \psi_{c}\cos \quad \varphi_{c}} \\{\cos \quad \psi_{c}\sin \quad \varphi_{c}} \\{\sin \quad \psi_{c}}\end{bmatrix}} & (2)\end{matrix}$

A functional representation of the coordinate transformation between thereference vector (r) and the vector associated with the solarconcentrator, i.e., the tracker vector (t), is shown in FIG. 7 and canbe expressed as follows: $\begin{matrix}{\begin{bmatrix}x_{t} \\y_{t} \\z_{t}\end{bmatrix} = {T_{\delta_{2}}T_{\lambda}T_{\delta_{1}}T_{\omega}T_{\gamma_{2}}{T_{\gamma_{1}}\begin{bmatrix}x_{r} \\y_{r} \\z_{r}\end{bmatrix}}}} & (3)\end{matrix}$

wherein the reference vector points at the sun.

The vector (X_(t), Y_(t), Z_(t)) defines the sun vector in the referencesystem of the solar concentrator. For a measurement system that includesa sun sensor mounted on the tracker and aligned with the centerline ofthe reflective surface, the sun sensor errors would be: $\begin{matrix}{{{Vertical}\quad {Error}} = {{VE} = {{\tan^{- 1}\left\lbrack \frac{z_{t}}{x_{t}} \right\rbrack} = {\tan^{- 1}\left\lbrack \frac{z_{e}}{x_{e}} \right\rbrack}}}} & (4) \\{{{Horizontal}\quad {Error}} = {{HE} = {{\tan^{- 1}\left\lbrack \frac{y_{t}}{x_{t}} \right\rbrack} = {\tan^{- 1}\left\lbrack \frac{y_{e}}{x_{e}} \right\rbrack}}}} & (5)\end{matrix}$

wherein the subscript “t” has been changed to “e” to denote the sunsensor error component.

As described above, the control system of the present invention thendetermines the contribution of each error source. By way of explanation,the derivation of the various equations that are utilized by the controlsystem of one embodiment to determine the relative contribution of eachof the error sources will hereinafter be described. The sun trackingmeasurement error transfer function relating the measurement errors tothe systematic error sources are obtained from Equation (3). Thetransformations shown in FIG. 7 are substituted into this equation withthe final result depicted in FIG. 8. The resulting equations are thensubstituted into Equations (4) and (5).

These equations are very complex and a complete analytical solution isvirtually impossible. However, by application of small angleapproximations to the error sources, these equations can be re-cast in aform that is solvable analytically for the approximate pertinent errorsources. Specifically, small angle approximations applied to thetransformation in FIG. 8, under the assumption that these errors areindeed small, are:

cos α≈1

sin α≈α

sin α sin β≈0

sin α sin β sin ξ≈0  (6)

to determine the following measurement error transfer functions:$\begin{matrix}\begin{matrix}{X_{t} = \quad {{X_{r}\left\lbrack {{\cos \quad {\lambda cos}\quad \omega} + {\sin \quad \lambda \quad \sin \quad \gamma_{1}} + {\sin \quad \lambda \quad \sin \quad \omega \quad \sin \quad \delta_{1}} - {\sin \quad \delta_{2}\sin \quad \omega}} \right\rbrack} +}} \\{\quad {{Y_{r}\left\lbrack {{\cos \quad {\lambda sin}\quad \omega} - {\sin \quad {\lambda cos}\quad {\omega sin}\quad \delta_{1}} - {\sin \quad \lambda \quad \sin \quad \gamma_{2}} + {\cos \quad {\omega sin\delta}_{2}}} \right\rbrack} +}} \\{\quad {Z_{r}\left\lbrack {{\sin \quad \lambda} + {\cos \quad {\lambda sin}\quad {\omega sin}\quad \gamma_{2}} - {\cos \quad \lambda \quad \cos \quad {\omega sin}\quad \gamma_{1}}} \right\rbrack}}\end{matrix} & (7) \\\begin{matrix}{Y_{t} = \quad {{X_{r}\left\lbrack {{{- \sin}\quad \omega} - {\sin \quad \delta_{2}\cos \quad {\lambda cos}\quad \omega}} \right\rbrack} +}} \\{\quad {{Y_{r}\left\lbrack {{\cos \quad \omega} - {\sin \quad \delta_{2}\cos \quad {\lambda sin}\quad \omega}} \right\rbrack} +}} \\{\quad {Z_{r}\left\lbrack {{\cos \quad {\omega sin}\quad \gamma_{2}} + {\sin \quad \delta_{1}} + {\sin \quad {\omega sin\gamma}_{1}} - {\sin \quad \delta_{2}\sin \quad \lambda}} \right\rbrack}}\end{matrix} & (8) \\\begin{matrix}{Z_{t} = \quad {{X_{r}\left\lbrack {{\cos \quad {\lambda sin\omega sin\delta}_{1}} + {\cos \quad {\lambda sin\gamma}_{1}} - {\sin \quad \lambda \quad \cos \quad \omega}} \right\rbrack} +}} \\{\quad {{Y_{r}\left\lbrack {{{- \cos}\quad {\lambda cos\omega sin\delta}_{1}} - {\cos \quad {\lambda sin\gamma}_{2}} - {\sin \quad {\lambda sin}\quad \omega}} \right\rbrack} +}} \\{\quad {Z_{r}\left\lbrack {{\cos \quad \lambda} + {\sin \quad {\lambda cos\omega sin}\quad \gamma_{1}} - {\sin \quad {\lambda sin\omega sin}\quad \gamma_{2}}} \right\rbrack}}\end{matrix} & (9)\end{matrix}$

wherein α, β and ξ denote arbitrary small angles measured in radians.Further, since ω=Φ_(c)+Φ_(e) and λ=Ψ_(c)−Ψ_(e)+g(Ψ)=e(Ψ)+r(Ψ)=Ψ_(c)+dΨ(see FIG. 7), the following identities result by selectively applyingthe above small angle approximations.

cos ω=cos(Φ_(c)+Φ_(e))=cos Φ_(c) cos Φ_(e)−sin Φ_(c) sin Φ_(e)=cosΦ_(c)−sin Φ_(c) sin Φ_(e)

sin ω=sin(Φ_(c)+Φ_(e))=sin(cos Φ_(c) cos Φ_(e)+sin Φ_(e) cos Φ_(c)=sinΦ_(c)+sin Φ_(e) cos Φ_(c)

cos λ=cos(Ψ_(c)+dΨ)=cos Ψ_(c) cos dΨ−sin Ψ_(c) sin dΨ=cos Ψ_(c)−sinΨ_(c) sin dΨ

sin λ=sin(Ψ_(c)+dΨ)=sin Ψ_(c) cos dΨ+sin dΨ cos Ψ_(c)=sin Ψ_(c)+sin dΨcos Ψ_(c)  (10)

For α denoting a small angle and applying Equations (6) and (10), it isnoted that:

sin α cos ω=sin α(cos Φ_(c)−sin Φ_(c) sin Φ_(e))=sin α cos Φ_(c)

sin α sin ω=sin α(sin Φ_(c)+sin Φ_(e) cos Φ_(c))=sin α sin Φ_(c)

sin α cos λ=sin α(cos Ψ_(c)−sin Ψ_(c) sin dΨ)=sin α cos Ψ_(c)

sin α sin λ=sin α(sin Ψ_(c)+sin dΨ cos Ψ_(c))=sin α sin Ψ_(c)  (11)

The command angles (Φ_(c) and Ψ_(c)) are computed using a sun model thatis assumed to have zero error relative to the real sun, e.g., zeromodeling error. The corresponding reference coordinates (X_(r), Y_(r)and Z_(r)), are defined with respect to the command angles as follows:$\begin{matrix}{\begin{bmatrix}X_{r} \\Y_{r} \\Z_{r}\end{bmatrix} = \begin{bmatrix}{\cos \quad \Phi_{c}\cos \quad \Psi_{c}} \\{\sin \quad \Phi_{c}\cos \quad \Psi_{c}} \\{\sin \quad \Psi_{c}}\end{bmatrix}} & (12)\end{matrix}$

By substituting Equations (6) and (10-12) in Equation (7) for X_(t), theexpression for X_(t) reduces to:

X_(t)=1  (13)

Similarly, Equations (8) and (9) for Y_(t) and Z_(t), respectivelyreduce to the following:

Y _(t)=sin Ψ_(c) sin Φ_(c) sin γ₁+sin Ψ_(c) cos Φ_(c) sin γ₂−cos Ψ_(c)sin Φ_(e)+sin Ψ_(c) sin δ₁−δ₂  (14)

Z _(t)=cos Φ_(c) sin γ₁−sin Φ_(c) sin γ₂ −dΨ  (15)

Equations (13)-(15) are the approximate transfer functions relating thereference system measurements and the individual platform error sources,i.e., the azimuth rotational tilt error γ₁ and γ₂, the elevationrotational tilt error Φ_(e) and δ₁, the reflective surfacenon-orthogonality error δ₂ and the elevation reference position errorΨ_(e). The unity value for the X measurement error is interpreted as noerror in the radial direction measured from the solar concentrator tothe sun due to systematic errors, while the Y component represents thehorizontal error and the Z component is the vertical error.

In order to completely define the total measurement error, the variousembodiments consider several additional error sources. These additionalerror sources include: (a) the gravity bending of the structure as it ispointed at the sun (g(Ψ_(c))), (b) the error in the elevation activatorreference position and the elevation actuator transfer function,(e(Ψ_(c))) and (c) the atmospheric refraction residual, i.e.,(r(Ψ_(c))). Including these additional terms, the following equationsrepresent the total measurement error both in the horizontal andvertical directions, respectively.

HE=sin Ψ_(c) sin Φ_(c) sin γ₁+sin Ψ_(c) cos Φ_(c) sin γ₂−cos Ψ_(c) sinΦ_(e)+sin Ψ_(c) sin δ₁−δ₂  (16)

VE=cos Φ_(c) sin γ₁−sin Φ_(c) sin γ₂−(−Ψ_(e)+g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c)))  (17)

The three terms (g, e, r), in Equation 17 are only a function of theelevation angle of the solar tracker. As such, the component of thevertical track error introduced by these three errors is symmetricalabout solar noon. Therefore:

g(Ψ_(c))|_(Φ) =g(Ψ_(c))|_(−Φ)

e(Ψ_(c))|_(Φ) =e(Ψ_(c))|_(−Φ)

r(Ψ_(c))|_(Φ) =r(Ψ_(c))|_(−Φ)  (18)

Note also that the first term in Equation (17) is also symmetric aboutdue South.

Now that the approximate measurement equations are known, thedetermination of analytical expressions for the errors can proceed.First, the two azimuth plane tilt parameters, γ₁ and γ₂, are determined.

Assume that the sun sensor measurements are taken over the day fromearly morning until late evening. Further assume that points are takensymmetrically about the South axis such that for every measurement atΦ_(c) there is a corresponding measurement taken at −Φ_(c). Using theseassumptions the vertical error measurement set is:

VE(Φ_(c))=cos Φ_(c) sin γ₁−sin Φ_(c) sin γ₂−(−Ψ_(e)+g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c)))

VE(−Φ_(c))=cos Φ_(c) sin γ₁+sin Φ_(c) sin γ₂−(−Ψ_(e)+g(Ψ_(c))+e(Ψ_(c)))+r(Ψ_(c)))  (19)

Subtracting these two equations and remembering the symmetry propertyfrom Equation (18), the following expression is obtained:

VE(Φ_(c))−VE(−Φ_(c))=−2 sin Φ_(c) sin γ₂  (20)

Note also that the gravity bending term, the elevation transfer functionterm and the atmospheric refraction term cancel. While each of theseterms is a function of the elevation angle, their functionality isweak—that is to say that for small differences in elevation angles,these terms are essentially the same and can be assumed to be equal,thus canceling when the two equations are subtracted. The resultingexpression for the tilt parameter, γ₂, from each pair of verticalsymmetrical measurements is: $\begin{matrix}{\gamma_{2} = {\sin^{- 1}\left\lbrack \frac{{{VE}\left( {- \Phi_{c}} \right)} - {{VE}\left( \Phi_{c} \right)}}{2\quad \sin \quad \Phi_{c}} \right\rbrack}} & (21)\end{matrix}$

The measurements indicated in Equation (21), e.g., the VE terms, containrandom noise due to the measurement instruments. To reduce the effectsof the noise, an estimate of the tilt parameter, γ₂ is obtained byaveraging the values of the tilt parameter from each of symmetricalpairs taken during the day as follows: $\begin{matrix}{\gamma_{2} = {\frac{1}{n}{\sum\limits_{i}\gamma_{2_{i}}}}} & (22)\end{matrix}$

wherein n is the total number of symmetric pairs and γ_(2i) are thevalues of the tilt parameter resulting from Equation (21) being appliedto each symmetric pair of measurements.

Similarly, the horizontal error measurement set is:

HE(Φ_(c))=(sin Ψ_(c))_(Φ) sin Φ_(c) sin γ₁+(sin Ψ_(c))_(Φ) cos Φ_(c) sinγ₂−(cos Ψ_(c))_(Φ) sin Φ_(e)+(sin Ψ_(c))_(Φ) sin δ₁−δ₂  (23)

HE(−Φ_(c))=−(sin Ψ_(c))_(−Φ) sin Φ_(c) sin γ₁+(sin Ψ_(c))_(−Φ) cos Φ_(c)sin γ₂−(cos Ψ_(c))_(−Φ) sin Φ_(e)+(sin Ψ_(c))_(−Φ) sin δ₁−δ₂  (24)

The notation for the elevation terms ( . . . )_(Φ) and ( . . . )_(−Φ′),denote which side of the solar concentrator meridian, e.g., which sideof due South, the horizontal measurements are made. The reason for thenotation is that while the azimuth measurements are symmetric about dueSouth, the corresponding elevation angles are not symmetrical and, ingeneral, are not equal. This notation allows for keeping track of thefact that the elevation angles are, in general, not the same on eachside of the tracker meridian.

Subtracting the two horizontal error measurements and keeping in mindthat the elevation terms will not cancel, the following is obtained:$\begin{matrix}\begin{matrix}{{{{HE}\left( \Phi_{c} \right)} - {{HE}\left( {- \Phi_{c}} \right)}} = \quad {{\left\lbrack {\sin \quad \Phi_{c}\sin \quad \gamma_{1}} \right\rbrack \left\lbrack {\left( {\sin \quad \Psi_{c}} \right)_{\Phi} + \left( {\sin \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack} +}} \\{\quad {{\left\lbrack {{\cos \quad \Phi_{c}\sin \quad \gamma_{2}} + {\sin \quad \delta_{1}}} \right\rbrack \left\lbrack {\left( {\sin \quad \Psi_{c}} \right)_{\Phi} - \left( {\sin \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack} -}} \\{\quad {\sin \quad {\Phi_{e}\left\lbrack {\left( {\cos \quad \Psi_{c}} \right)_{\Phi} - \left( {\cos \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack}}}\end{matrix} & (25)\end{matrix}$

Solving Equation (25) for the tilt parameter, γ₁: $\begin{matrix}\begin{matrix}{\gamma_{1} = \quad {\sin^{- 1}\left\{ {\frac{{{HE}\left( \Phi_{c} \right)} - {{HE}\left( {- \Phi_{c}} \right)}}{\sin \quad {\Phi_{c}\left\lbrack {\left( {\sin \quad \Psi_{c}} \right)_{\Phi} + \left( {\sin \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack}} -} \right.}} \\{\quad {\frac{\left\lbrack {{\cos \quad \Phi_{c}\sin \quad \gamma_{2}} + {\sin \quad \delta_{1}}} \right\rbrack \left\lbrack {\left( {\sin \quad \Psi_{c}} \right)_{\Phi} - \left( {\sin \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack}{\sin \quad {\Phi_{c}\left\lbrack {\left( {\sin \quad \Psi_{c}} \right)_{\Phi} + \left( {\sin \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack}} +}} \\\left. \quad \frac{\sin \quad {\Phi_{e}\left\lbrack {\left( {\cos \quad \Psi_{c}} \right)_{\Phi} - \left( {\cos \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack}}{\sin \quad {\Phi_{c}\left\lbrack {\left( {\sin \quad \Psi_{c}} \right)_{\Phi} + \left( {\sin \quad \Psi_{c}} \right)_{- \Phi}} \right\rbrack}} \right\}\end{matrix} & (26)\end{matrix}$

The appearance of a second unknown parameter, δ₁, in the second term ofEquation (26) precludes a general analytical solution for the tiltparameter. However, taking measurements at a time of year when thediurnal motion of the sun with respect to the solar concentratorproduces little variation in the elevation angle, for symmetric azimuthcommands, makes the estimate of the tilt parameter possible.Specifically, if measurements are taken at or near either the Summer orWinter Solstice, the apparent motion of the sun with respect to thesolar concentrator is approximately symmetrical about the solarconcentrator meridian. As such, the values of the elevation command(Ψ_(c)) for each symmetric azimuth command pair are nearly equal. Whenmeasurements are taken at this time of year, then, the second and thirdterms of Equation (26) are nearly zero. The equation for the tiltparameter then reduces to: $\begin{matrix}{\gamma_{1} = {\sin^{- 1}\left\{ \frac{{{HE}\left( \Phi_{c} \right)} - {{HE}\left( {- \Phi_{c}} \right)}}{2\sin \quad \Phi_{c}\sin \quad \Psi_{c}} \right\}}} & (27)\end{matrix}$

At the Vernal and Autumnal Equinoxes, the daily change of the sunelevation position, for symmetric azimuths, during the day is at itsmaximum, e.g., the difference in the sun elevation between early morningand late afternoon can be as much as 5 mrad. This difference decreasesas the sample point pairs are taken closer to solar noon. But, even whenthe symmetric azimuth samples are taken in the early morning and lateafternoon, the possible difference of 5 mrad. between the correspondingelevation angles will make the second and third terms of Equation (26)second order and can thus be neglected. It is expected, then, thatEquation (27) will produce a very good estimate of the second tiltparameter.

As in the case for the previous tilt parameter, γ₂, to get a finalestimate of γ₁ and reduce the effects of measurement noise, a simpleaverage over the set of tilt parameters obtained from each of thesymmetrical pairs of measurements, e.g., the HE terms, taken during theday is performed as follows: $\begin{matrix}{\gamma_{1} = {\frac{1}{n}{\sum\limits_{i}\gamma_{1_{i}}}}} & (28)\end{matrix}$

wherein n is the total number of symmetric pairs and γ_(1i) are thevalues of the tilt parameter resulting from Equation (27) being appliedto each symmetric pair of measurements.

Now that the azimuth tilt plane is known, an estimate of the remainingerror terms in Equation (17) can proceed.

Solving for the unknown error terms in Equation (17) produces:

r(Ψ_(c))+e(Ψ_(c))+g(Ψ_(c))−Ψ_(e) =−VE+cos Φ_(c) sin γ₁−sin Φ_(c) sinγ₂  (29)

The first, second and third terms on the left hand side of Equation (29)are a function of the elevation angle and the last term is a constant.The right hand side of Equation (29) can be curve fit with a polynomialusing the measured values of VE, the commended azimuth angle Φ_(c), andthe values of δ₁ and δ₂ determined from Equations (22) and (28). In thisregard, the curve fit is a quadratic fit of the measurements as afunction of elevation angle. This curve fit can then be used to correctfor the combined errors on the left hand side of the equation. Note thatit is difficult but not impossible to solve for the individual errorterms if desired. For example, an inclinometer can be used to measurethe gravitational bending. If the elevation linear actuator, i.e., theelevation jack, is set at the reference point accurately, then e(Ψ_(c))should be zero. The measurements are also preferably taken at highelevation angles so that refraction errors and the error residual,r(Ψ_(c)) are minimized. The remaining unknown error, Ψ_(e), can then befound easily. However, it is not necessary to determine the individualerrors since only the combined errors are required for the controllogic.

The remaining error parameters, (δ₁, δ₂, Φ_(e)) can be found fromEquation (16). Solving this equation for the combined errors gives:

HE−sin Ψ_(c) sin Φ_(c) sin γ₁−sin Ψ_(c) cos Φ_(c) sin γ₂=−cos Ψ_(c) sinΦ_(e)+sin Ψ_(c) sin δ₁−δ₂  (30)

Now define the following variables with the subscript “i” refers to the“ith” measurement:

M_(i) is the left hand side of Equation (30)

A=sin δ₁

B=sin Φ_(e)

C=δ ₂  (31)

Equation (30) can be rewritten as:

M _(i) =A sin Ψ_(i) −B cos−Ψ_(i) −C  (32)

Let the square error function, V, be: $\begin{matrix}{V = {\sum\limits_{i = 1}^{n}\left( {M_{i} - {A\quad \sin \quad \Psi_{i}} + {B\quad \cos \quad \Psi_{i}} + C} \right)^{2}}} & (33)\end{matrix}$

The curve fit coefficients (A, B and C) that relate the angle, Ψ_(i), tothe measurements, M_(i), are then determined by taking the partialderivatives with respect to each of the coefficents (A, B and C),setting them to zero and solving simultaneously for them.$\begin{matrix}{\frac{\partial V}{\partial A} = {0 = {2{\sum\limits_{i = 1}^{n}{\left( {M_{i} - {A\quad \sin \quad \Psi_{i}} + {B\quad \cos \quad \Psi_{i}} + C} \right)\left( {{- \sin}\quad \Psi_{i}} \right)}}}}} & (34)\end{matrix}$

Combining and rearranging gives: $\begin{matrix}{{{A{\sum\limits_{i = 1}^{n}{\sin^{2}\Psi_{i}}}} - {B{\sum\limits_{i = 1}^{n}{\sin \quad \Psi_{i}\cos \quad \Psi_{i}}}} - {C{\sum\limits_{i = 1}^{n}{\sin \quad \Psi_{i}}}}} = {\sum\limits_{i = 1}^{n}{M_{i}\sin \quad \Psi_{i}}}} & (35)\end{matrix}$

Similarly, the partial derivatives with respect to B and C yield thefollowing equations, respectively: $\begin{matrix}{{{A{\sum\limits_{i = 1}^{n}{\sin \quad \Psi_{i}\cos \quad \Psi_{i}}}} - {B{\sum\limits_{i = 1}^{n}{\cos^{2}\Psi_{i}}}} - {C{\sum\limits_{i = 1}^{n}{\cos \quad \Psi_{i}}}}} = {\sum\limits_{i = 1}^{n}{M_{i}\cos \quad \Psi_{i}}}} & (36) \\{{{A\quad {\sum\limits_{i = 1}^{n}{\sin \quad \Psi_{i}}}} - {B{\sum\limits_{i = 1}^{n}{\cos \quad \Psi_{i}}}} - {Cn}} = {\sum\limits_{i = 1}^{n}M_{i}}} & (37)\end{matrix}$

To simplify the notation, make the following substitutions.$\begin{matrix}\begin{matrix}{z = {\sum\limits_{i = 1}^{n}{\sin^{2}\Psi_{i}}}} & {y = {\sum\limits_{i = 1}^{n}{\sin \quad \Psi_{i}\cos \quad \Psi_{i}}}} & {x = {\sum\limits_{i = 1}^{n}{\sin \quad \Psi_{i}}}} \\{w = {\sum\limits_{i = 1}^{n}{M_{i}\sin \quad \Psi_{i}}}} & {v = {\sum\limits_{i = 1}^{n}{\cos^{2}\Psi_{i}}}} & {u = {\sum\limits_{i = 1}^{n}{\cos \quad \Psi_{i}}}} \\{t = {\sum\limits_{i = 1}^{n}{M_{i}\cos \quad \Psi_{i}}}} & {s = {\sum\limits_{i = 1}^{n}M_{i}}} & \quad\end{matrix} & (38)\end{matrix}$

Equations (36)-(38) can then be rewritten with the appropriatesubstitutions as follows:

Az−By−Cx=w  (39)

Ay−Bv−Cu=t  (40)

Ax−Bu−Cn=s  (41)

There are at least two ways to solve for the coefficients A, B and C.The first is to cast the set of equations as a matrix equation and solvefor the coefficients using matrix algebra. Another way is by directalgebraic manipulation of the individual equations. The latter way isused here since the algebra and the matrix inversion associated with theformer method require more complex mathematical manipulations.

Begin by multiplying Equation (39) by (−u), then multiply Equation (40)by (x) and add the two resulting equations together: $\begin{matrix}{{{{- {Auz}} + {Buy} + {Cux}} = {- {wu}}}\underset{\_}{{{Ayx} - {Bvs} - {Cux}} = {tx}}{{{A\left( {{yx} - {uz}} \right)} + {B\left( {{uy} - {vx}} \right)}} = {{tx} - {wu}}}} & (42)\end{matrix}$

Next, multiply Equation (39) by (−n), multiply Equation (41) by (x) andadd the two resulting equations together: $\begin{matrix}{{{{- {Anz}} + {ny} + {Cnx}} = {- {wn}}}\underset{\_}{{{Ax}^{2} - {Bux} - {Cnx}} = {sx}}{{{A\left( {x^{2} - {nz}} \right)} + {B\left( {{ny} - {ux}} \right)}} = {{sx} - {wn}}}} & (43)\end{matrix}$

Now, multiply Equation (42) by −(ny−ux), multiply Equation (43) by(uy−vx) and add the two resulting equations together: $\begin{matrix}{{{{{- {A\left( {{yx} - {uz}} \right)}}\left( {{ny} - {ux}} \right)} - {{B\left( {{uy} - {vx}} \right)}\left( {{ny} - {ux}} \right)}} = {{- \left( {{tx} - {wu}} \right)}\left( {{ny} - {ux}} \right)}}\underset{\_}{{{{A\left( {x^{2} - {nz}} \right)}\left( {{uy} - {vx}} \right)} + {{B\left( {{uy} - {vx}} \right)}\left( {{ny} - {ux}} \right)}} = {\left( {{sx} - {wn}} \right)\left( {{uy} - {vx}} \right)}}{{A\left\lfloor {{\left( {x^{2} - {nz}} \right)\left( {{uy} - {vx}} \right)} - {\left( {{yx} - {uz}} \right)\left( {{ny} - {ux}} \right)}} \right\rfloor} = {{\left( {{sx} - {wn}} \right)\left( {{uy} - {vx}} \right)} - {\left( {{tx} - {wu}} \right)\left( {{ny} - {ux}} \right)}}}} & (44)\end{matrix}$

Solving for A: $\begin{matrix}{A = \frac{{\left( {{sx} - {wn}} \right)\left( {{uy} - {vx}} \right)} - {\left( {{tx} - {wu}} \right)\left( {{ny} - {ux}} \right)}}{{\left( {x^{2} - {nz}} \right)\left( {{uy} - {vx}} \right)} - {\left( {{yx} - {uz}} \right)\left( {{ny} - {ux}} \right)}}} & (45)\end{matrix}$

Substitute A into Equation (43) and solve for B: $\begin{matrix}{B = \frac{{- {A\left( {x^{2} - {nz}} \right)}} + \left( {{sx} - {wn}} \right)}{\left( {{ny} - {ux}} \right)}} & (46)\end{matrix}$

Now substitute for A and B in Equation (39) and solve for C:$\begin{matrix}{C = \frac{{Az} - {By} - w}{x}} & (47)\end{matrix}$

The estimates of the three remaining errors are found as follows fromthe curve fit coefficients:

δ₁=sin⁻¹(A)

Φ_(e)=sin⁻¹(B)

δ₂ =C  (48)

The control system 27 then determines the elevation and azimuth commandsbased at least partially upon the respective errors generated by thevariety of error sources. In particular, the control system determinesthe azimuth command angle Φ_(c) and the elevation command angle Ψ_(c) inorder to align the centerline 15 defined by the reflective surface 12 ofthe solar concentrator 10 and the sun reference vector by taking intoaccount or compensating for the effects of each of the variety of errorsources described above. By way of explanation, the derivation of thevarious equations that relate the respective contributions of thevarious error sources to the elevation and azimuth commands determinedby the control system of the present invention will be hereinafterdescribed. The development of the equations begins by solving for thecommand angles from Equation (3). Multiply this equation by the inverseof the first four transition matrices and note also that the desiredtracking vector is aligned along the X_(t) axis. $\begin{matrix}{{T_{\omega}^{- 1}T_{\delta_{1}}^{- 1}T_{\lambda}^{- 1}{T_{\delta_{2}}^{- 1}\begin{bmatrix}1 \\0 \\0\end{bmatrix}}} = {T_{\gamma_{2}}{T_{\gamma_{1}}\begin{bmatrix}x_{r} \\y_{r} \\z_{r}\end{bmatrix}}}} & (49)\end{matrix}$

Now, progressing through each multiplication step-by-step, the followingis obtained.

The first multiplication: $\begin{matrix}{{T_{\delta_{2}}^{- 1}\begin{bmatrix}1 \\0 \\0\end{bmatrix}} = {{\begin{bmatrix}{\cos \quad \delta_{2}} & {{- \sin}\quad \delta_{2}} & 0 \\{\sin \quad \delta_{2}} & {\cos \quad \delta_{2}} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 \\0 \\0\end{bmatrix}} = \begin{bmatrix}{\cos \quad \delta_{2}} \\{\sin \quad \delta_{2}} \\0\end{bmatrix}}} & (50)\end{matrix}$

The next multiplication yields: $\begin{matrix}{{T_{\lambda}^{- 1}{T_{\delta_{2}}^{- 1}\begin{bmatrix}1 \\0 \\0\end{bmatrix}}} = {{\begin{bmatrix}{\cos \quad \lambda} & 0 & {{- \sin}\quad \lambda} \\0 & 1 & 0 \\{\sin \quad \lambda} & 0 & {\cos \quad \lambda}\end{bmatrix}\begin{bmatrix}{\cos \quad \delta_{2}} \\{\sin \quad \delta_{2}} \\0\end{bmatrix}} = \begin{bmatrix}{\cos \quad \lambda \quad \cos \quad \delta_{2}} \\{\sin \quad \delta_{2}} \\{\sin \quad {\lambda cos}\quad \delta_{2}}\end{bmatrix}}} & (51)\end{matrix}$

The third multiplication yields: $\begin{matrix}\begin{matrix}{{T_{\delta_{1}}^{- 1}T_{\lambda}^{- 1}{T_{\delta_{2}}^{- 1}\begin{bmatrix}1 \\0 \\0\end{bmatrix}}} = \quad {\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \quad \delta_{1}} & {{- \sin}\quad \delta_{1}} \\0 & {\sin \quad \delta_{1}} & {\cos \quad \delta_{1}}\end{bmatrix}\begin{bmatrix}{\cos \quad {\lambda cos}\quad \delta_{2}} \\{\sin \quad \delta_{2}} \\{\sin \quad {\lambda cos}\quad \delta_{2}}\end{bmatrix}}} \\{= \quad \begin{bmatrix}{\cos \quad {\lambda cos}\quad \delta_{2}} \\{{\cos \quad \delta_{1}\sin \quad \delta_{2}} - {\sin \quad \lambda \quad \sin \quad \delta_{1}\cos \quad \delta_{2}}} \\{{\sin \quad \delta_{1}\sin \quad \delta_{2}} + {\sin \quad \lambda \quad \cos \quad \delta_{1}\cos \quad \delta_{2}}}\end{bmatrix}}\end{matrix} & (52)\end{matrix}$

The final multiplication results in the following for the left hand sideof Equation (49): $\begin{matrix}{\begin{matrix}{{T_{\omega}^{- 1}T_{\delta_{1}}^{- 1}T_{\lambda}^{- 1}{T_{\delta_{2}}^{- 1}\begin{bmatrix}1 \\0 \\0\end{bmatrix}}} =}\end{matrix}\begin{matrix}{{\begin{bmatrix}{\cos \quad \omega} & {{- \sin}\quad \omega} & 0 \\{\sin \quad \omega} & {\cos \quad \omega} & 0 \\0 & 0 & 1\end{bmatrix}\begin{matrix}\begin{bmatrix}{\cos \quad \lambda \quad \cos \quad \delta_{2}} \\{{\cos \quad \delta_{1}\sin \quad \delta_{2}} - {\sin \quad \lambda \quad \sin \quad \delta_{1}\cos \quad \delta_{2}}} \\{{\sin \quad \delta_{1}\sin \quad \delta_{2}} + {\sin \quad {\lambda cos}\quad \delta_{1}\cos \quad \delta_{2}}}\end{bmatrix}\end{matrix}} =}\end{matrix}\begin{matrix}\begin{bmatrix}{{\cos \quad \omega \quad \cos \quad \lambda \quad \cos \quad \delta_{2}} - {\sin \quad \omega \quad \cos \quad \delta_{1}\sin \quad \delta_{2}} + {\sin \quad {\omega sin}\quad \lambda \quad \sin \quad \delta_{1}\cos \quad \delta_{2}}} \\{{\sin \quad {\omega cos}\quad {\lambda cos}\quad \delta_{2}} + {\cos \quad \omega \quad \cos \quad \delta_{1}\sin \quad \delta_{2}} - {\cos \quad \omega \quad \sin \quad {\lambda sin}\quad \delta_{1}\cos \quad \delta_{2}}} \\{{\sin \quad \delta_{1}\sin \quad \delta_{2}} + {\sin \quad \lambda \quad \cos \quad \delta_{1}\cos \quad \delta_{2}}}\end{bmatrix}\end{matrix}} & (53)\end{matrix}$

The next step is to expand the right hand side of Equation (49) andsubstitute Equation (1) for the reference vector. $\begin{matrix}{\begin{matrix}{{T_{\gamma_{2}}{T_{\gamma_{1}}\begin{bmatrix}x_{r} \\y_{r} \\z_{r}\end{bmatrix}}} = {{{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \quad \gamma_{2}} & {\sin \quad \gamma_{2}} \\0 & {{- \sin}\quad \gamma_{2}} & {\cos \quad \gamma_{2}}\end{bmatrix}\begin{bmatrix}{\cos \quad \gamma_{1}} & 0 & {{- \sin}\quad \gamma_{1}} \\0 & 1 & 0 \\{\sin \quad \gamma_{1}} & 0 & {\cos \quad \gamma_{1}}\end{bmatrix}}\begin{bmatrix}x_{r} \\y_{r} \\z_{r}\end{bmatrix}} =}}\end{matrix}\begin{matrix}{{\begin{bmatrix}{\cos \quad \gamma_{1}} & 0 & {{- \sin}\quad \gamma_{1}} \\{\sin \quad \gamma_{1}\sin \quad \gamma_{2}} & {\cos \quad \gamma_{2}} & {\cos \quad \gamma_{1}\sin \quad \gamma_{2}} \\{\sin \quad \gamma_{1}\cos \quad \gamma_{2}} & {{- \sin}\quad \gamma_{2}} & {\cos \quad \gamma_{1}\cos \quad \gamma_{2}}\end{bmatrix}\begin{bmatrix}{\cos \quad \Psi_{r}\cos \quad \Phi_{r}} \\{\cos \quad \Psi_{r}\sin \quad \Phi_{r}} \\{\sin \quad \Psi_{r}}\end{bmatrix}} =}\end{matrix}\begin{matrix}\begin{bmatrix}{{\cos \quad \gamma_{1}\cos \quad \Psi_{r}\cos \quad \Phi_{r}} - {\sin \quad \gamma_{1}\sin \quad \Psi_{r}}} \\{{\sin \quad \gamma_{1}\sin \quad \gamma_{2}\cos \quad \Psi_{r}\cos \quad \Phi_{r}} + {\cos \quad \gamma_{2}\cos \quad \Psi_{r}\sin \quad \Phi_{r}} + {\sin \quad \Psi_{r}\cos \quad \gamma_{1}\sin \quad \gamma_{2}}} \\{{\sin \quad \gamma_{1}\cos \quad \Psi_{r}\cos \quad \Phi_{r}\cos \quad \gamma_{2}} - {\sin \quad \gamma_{2}\cos \quad \Psi_{r}\sin \quad \Phi_{r}} + {\cos \quad \gamma_{1}\cos \quad \gamma_{2}\sin \quad \Psi_{r}}}\end{bmatrix}\end{matrix}} & (54)\end{matrix}$

The approach to finding the elevation command angle, Ψ_(c), is to equatethe third component of Equation (53) to the third component of Equation(54), solve for λ and, using the definition λ=Ψ_(c)+dΨ, solve for Ψ_(c).$\begin{matrix}\begin{matrix}{{{\sin \quad \delta_{1}\sin \quad \delta_{2}} + {\sin \quad {\lambda cos}\quad \delta_{1}\cos \quad \delta_{2}}} = \quad {{\sin \quad \gamma_{1}\cos \quad \Psi_{r}\quad \cos \quad \Phi_{r}\cos \quad \gamma_{2}} -}} \\{\quad {{\sin \quad \gamma_{2}\cos \quad \Psi_{r}\sin \quad \Phi_{r}} + {\cos \quad \gamma_{1}\cos \quad \gamma_{2}\sin \quad \Psi_{r}}}}\end{matrix} & (55) \\{{\sin \quad \lambda} = {\frac{\begin{matrix}{{\sin \quad \gamma_{1}\cos \quad \Psi_{r}\cos \quad \Phi_{r}\cos \quad \gamma_{2}} - {\sin \quad \gamma_{2}\cos \quad \Psi_{r}\sin \quad \Phi_{r}} +} \\{{\cos \quad \gamma_{1}\cos \quad \gamma_{2}\sin \quad \Psi_{r}} - {\sin \quad \delta_{1}\sin \quad \delta_{2}}}\end{matrix}}{\cos \quad \delta_{1}\cos \quad \delta_{2}} = {\sin \left( {\Psi_{c} + {d\quad \Psi}} \right)}}} & (56) \\{{Solving}\quad {for}\quad {the}\quad {elevation}\quad {command}\quad {angle}\quad \Psi_{c}\quad {yields}\text{:~~~~~~~~~~~~~~~~~~~~}} & \quad \\{\Psi_{c} = {{\sin^{- 1}\left\lbrack {\frac{{\sin \quad \gamma_{1}\cos \quad \Psi_{r}\cos \quad \Phi_{r}\cos \quad \gamma_{2}} - {\sin \quad \gamma_{2}\cos \quad \Psi_{r}\sin \quad \Phi_{r}}}{\cos \quad \delta_{1}\cos \quad \delta_{2}} + \frac{{\cos \quad \gamma_{1}\cos \quad \gamma_{2}\sin \quad \Psi_{r}} - {\sin \quad \delta_{1}\sin \quad \delta_{2}}}{\cos \quad \delta_{1}\cos \quad \delta_{2}}} \right\rbrack} - {d\quad \Psi}}} & (57)\end{matrix}$

Applying the small angle approximations provided earlier (Equation (6)),the equation defining the elevation command angle Ψ_(c), simplifies tothe following:

Ψ_(c)=sin⁻¹(sin γ₁ cos Ψ_(r) cos Φ_(r)−sin γ₂ cos Ψ_(r) sin Φ_(r)+sinΨ_(r))−dΨ  (58)

To solve for the azimuth command angle Φ_(c) including the error sourceestimates, the following definitions are made to facilitate thederivation and notation.

A=cos λ cos δ₂

B=cos δ₁ sin δ₂−sin δ₁ cos δ₂ sin λ

C=cos γ₁ cos Ψ_(r) cos Φ_(r)−sin γ₁ sin Ψ_(r)

D=sin γ₁ sin γ₂ cos Ψ_(r) cos Φ_(r)+cos γ₂ cos Ψ_(r) sin Φ_(r)+sin Ψ_(r)cos γ₁ sin γ₂  (59)

Substituting these definitions into Equations (53) and (54) and equatingthe first and second components of these equations yields the following:

A cos ω−B sin ω=C

B cos ω+A sin ω=D  (60)

Since ω=Φ_(c)+Φ_(e), the solution for the azimuth command angle becomes,from Equation (60): $\begin{matrix}{~{\Phi_{c} = {{\cos^{- 1}\left\lbrack \frac{{A\quad C} + {BD}}{A^{2} + B^{2}} \right\rbrack} - \Phi_{e}}}} & (61)\end{matrix}$

Note that the azimuth command contains λ on the right-hand side which isa function of the elevation command angle Ψ_(c) from Equation (57). Theelevation command angle must therefore be solved for first. Also,because of this complexity, the equations for the elevation and azimuthcommands cannot be solved separately.

Based upon the elevation and azimuth angle at which the sun is locatedas determined by the ephemeris equations and as a result of the relativecontributions of the various error sources determined in the mannerdescribed above based upon the vertical and horizontal errors determinedby the measurement system 26, the elevation and azimuth command anglescan be readily determined from Equations (58) and (61), respectively. Assuch, the control system 27 of the present system includes an outputsection for providing signals representative of the elevation commandand the azimuth command and, more particularly, the elevation andazimuth command angles, in order to command the solar concentrator 10 toa position that compensates for the vertical and horizontal errorsdetected by the measurement system.

Correspondingly, the apparatus of the present invention preferablyincludes a positioning mechanism, such as the motor 32 and theassociated azimuthal and elevational drive mechanisms 34, 36 forreceiving the signals provided by the output section of the controlsystem 27 and for positioning the solar concentrator 10 based upon theelevation and azimuth commands. As such, the solar concentrator can bepositioned such that the centerline 15 defined by the reflective surface12 is aligned with the sun reference vector, thereby optimizing thecollection of solar energy by the solar concentrator. By repeating thisprocess of fine-tuning the position of the solar concentrator tocompensate for each of the various error sources at a plurality of timesthroughout the day, the solar concentrator can remain trained upon theposition of the sun even as the position of the sun and thecorresponding position of the solar concentrator change.

As described above, the control system 27 includes a processing element28. Typically, the processing element comprises one or more processorsor other computing elements or devices for performing the functionsdescribed above. However, the processing element of the control systemcan include any type of processing element known to those skilled in theart that are capable of performing the functions described above. Whilethe functions provided by the processing element can be provided byhardware, such as by an application specific integrated circuit (ASIC)or the like, the functionality is typically provided by a computerprogram product embodied within the processing element or stored withinthe memory device associated with the processing element and adapted foroperation by the processing element. As such, one embodiment of thepresent invention provides a computer program product having acomputer-readable storage medium, such as a read only memory device, arandom access memory device or the like, having computer-readableprogram code provided therein for performing the various functionsdescribed above in conjunction with the processing element of thecontrol system.

By way of example of the method and apparatus of the present invention,a simulation that was conducted during the summer solstice ishereinafter described. As explained in more detail below, the simulationincluded the following steps: (1) the azimuth and elevation commandangles were input, (2) the actual or true errors applicable to theheliostat were then defined including any elevation angle dependency,(3) the vertical and horizontal error measurements taken at eachazimuth/elevation setting of the heliostat during the day including theactual error in data obtained during the summer solstice were thendefined, (4) the errors were then estimated in the same manner as ifactual measurement data were available utilizing the vertical andhorizontal error measurements defined in the preceding step, (5) theroot mean square (RMS) error over the day was then determined, and (6)the azimuth and elevation commands were modified based upon the RMSerror and the resulting effect on the errors is determined.

The initial step is to obtain the azimuth and elevation command anglesΦ_(c) and Ψ_(c) for the summer solstice as depicted in FIG. 10, such asfrom a data file. The data defining the azimuth and elevation commandangles are representative data from a solar ephemeris model defining theazimuth and elevation angles to the sun from the heliostat during thesolar day. As shown, the azimuth command angles are symmetrical aboutdue south from the heliostat meridian. There are twenty-four symmetricpairs of azimuth/elevation data used in this example. Thus, the totalnumber of measurements k in this simulation is forty-eight, while thenumber of symmetric pairs n is twenty-four. Moreover, the subscripts iand j are defined as i=1 . . . k and j=1 . . . n.

The second step in this example is to define the true errors applicableto the heliostat as described above and defined below. The error modelis also defined, as denoted by the variable “noise”, representing theerror associated with the measurement sensor.

Noise =0.5 mrad${{{uniformly}\quad {distributed}\quad {between}}\quad + \text{/} - \frac{noise}{2}} = {0.25\quad {mrad}}$

δ₁=−2 mrad δ₂=2 mrad

Φ_(e)=5 mrad γ₁=5 mrad γ₂=5 mrad

Ψ_(e)=1 mrad

The data defining the gravitational residue error g, the elevationtransfer function error e and the error due to atmospheric refraction rare also obtained from data files. The data is only dependent upon theelevation angle and, once obtained, the data is applied symmetricallyabout the heliostat meridian to provide the data for use as a functionof azimuth angle. For purposes of this example, the gravitationalresidue error g, the elevation transfer function error e and the errordue to atmospheric refraction r are collectively termed system elevationerrors and are depicted in FIG. 11.

As described above, dΨ_(tj)=g_(j)+e_(j)+r_(j)−Ψ_(e) can therefore alsobe defined as the combined true elevation error shown in FIG. 12,wherein Ψ_(e) is the true error defined above. As defined below,dΨ_(truej) is the combination of the gravitational residue error g, theelevation transfer function error e and the error due to atmosphericrefraction r as well as the elevation reference error. For negativeazimuth command angles, dΨ_(truej) is defined as: dΨ_(truej)=dΨ_(tj).For positive azimuth command angles, dΨ_(truej) is defined as: dΨ_(true)_(n+j) =dΨ_(truej) _(n−j+1) . dΨ_(truei) is termed the true elevationerror and is also depicted in FIG. 13.

The third step is a definition of equations used to calculate thevertical and horizontal measurement errors. As defined above byequations (4) and (5), the vertical and horizontal measurement errorsare defined as follows:${{VEm}\left( {Z_{t},X_{t}} \right)} = {a\quad {\tan \left\lbrack \frac{Z_{t}}{X_{t}} \right\rbrack}}$${{HEm}\left( {Y_{t},X_{t}} \right)} = {a\quad {\tan \left\lbrack \frac{Y_{t}}{X_{t}} \right\rbrack}}$

This step also includes the calculation of the measurement errors usingthe heliostat and system errors defined in the prior step and thepointing commands from the initial step. These calculations representthe errors that would have been measured during the day for thissimulation. In particular, the vertical measurement errors aredetermined utilizing the transformation matrix of FIG. 8 includingmeasurement noise. The true value of the vertical measurement error withno noise is defined below:

VE _(Φt) _(i) =VEm(Z _(ti) ,X _(ti))

In addition, the measured value of the vertical measurement error whichincludes noise is defined as follows:

VE _(Φi) =VE _(Φti)+noise(rnd(1)−0.5)

wherein (rnd(1)−0.5) is a function that produces a uniformly distributedrandom number between −0.5 and 0.5.

In each of these equations, the variables ω_(i), λ_(i), Z_(ti) andX_(ti) are defined as follows:

ω_(i)=φ_(ci)+Φ_(e)

λ_(i)=ψ_(ci) +dΨ _(truei)

Z _(ti) =T ₃₁(δ₁,δ₂,γ₁,γ₂,λ_(i),

ω_(i))cos(ψ_(ci))cos(φ_(ci))+T ₃₂

(δ₁,δ₂,γ₁,γ₂,λ_(i),ω_(i))

cos(ψ_(ci))sin(φ_(ci))+T ₃₃

(δ₁,δ₂,γ₁,γ₂λ_(i),

ω_(i))sin(ψ_(ci))

X _(ti) =T ₁₁(δ₁,δ₂,γ₁,γ₂λ_(i),ω_(i))

cos(ψ_(ci))cos(φ_(ci))+T ₁₂

(δ₁,δ₂,γ₁,γ₂,λ_(i),

ω_(i))cos(ψ_(ci))sin(φ_(ci))+T ₁₃

(δ₁,δ₂,γ₁,γ₂,λ_(i),

ω_(i))sin(ψ_(ci))

The vertical measurement errors are depicted in FIG. 14 in which thesolid line represents the true vertical measurement error and the squareboxes represent the measured values of the vertical measurement error.

Similarly, both the true and measured values of the horizontalmeasurement error are determined as follows:

HE _(Φti) =HEm(Y _(ti) ,X _(ti))

HE _(Φi) =HE _(Φ+i)+noise(rnd(1)−0.5)

In this regard, variable Y_(ti) is defined as follows:

X _(ti) =T ₁₁(δ₁,δ₂,γ₁,

γ₂,λ_(i),ω_(i))cos(ψ_(ci))

cos(φ_(ci))+T ₁₂(δ₁,δ₂,

γ₁,γ₂,λ_(i),ω_(i))

cos(ψ_(ci))sin(φ_(ci))+T ₁₃

(δ₁,δ₂,γ₁,γ₂,λ_(i),

ω_(i))sin(ψ_(ci))

Similar to the true and measured values of the vertical measurementerror, the true measured values of the horizontal measurement error aredepicted in FIG. 15 in which the solid line represents the true value ofthe horizontal measurement error and the square boxes represent themeasured values of the horizontal measured error.

The data from the prior step representing the collective vertical andhorizontal measurement errors is then processed to determine theindividual error terms. Based on these error terms, the commands used onsubsequent days can be refined to reduce the measurement errors and thusincrease the efficiency of the heliostat. The first tilt parameter isestimated as follows:$\gamma_{1{ej}} = {a\quad {\sin \left\lbrack \frac{{HE}_{{\varphi \quad k} - j + 1} - {HE}_{\varphi \quad j}}{2{{\sin \left( \varphi_{cj} \right)}}{\sin \left( \psi_{cj} \right)}} \right\rbrack}}$

The mean value of the first tilt parameter can therefore be defined as$\gamma_{1{est}} = {\frac{1}{n}{\sum\limits_{j}\gamma_{1{ej}}}}$

Based upon the horizontal measurement errors depicted in FIG. 15 and theazimuth and elevation command angles depicted in FIG. 10, the first tiltparameter is 4.86 mrad. To determine the error that arises as a resultof utilizing the mean value of the first parameter, the mean estimationerror can be determined as follows:

me _(γ1)=γ_(1est)−γ₁

In this instance, the mean estimation error is equal to −0.13 mrad. Inaddition, the percent estimation error can be defined as follows:$\frac{{me}_{\gamma 1}}{\gamma_{1}} = {{- 2.6}\%}$

Similarly, the second tilt parameter can be defined as follows:$\lambda_{2{ej}} = {a\quad {\sin \left\lbrack \frac{{VE}_{\varphi \quad j} - {VE}_{{\varphi \quad k} - j + 1}}{2{{\sin \left( \varphi_{cj} \right)}}} \right\rbrack}}$

The mean value of the second tilt parameter can then be defined as:$\gamma_{2{est}} = {\frac{1}{n}{\sum\limits_{j}\gamma_{2{ej}}}}$

For the vertical measurement error depicted in FIG. 14, and the azimuthand elevation command angles of FIG. 10, the mean value of the secondtilt parameter is 5.00 mrad. In the same manner as defined above, themean estimation error is 0.003 mrad and the percent estimation error is0.05%.

The combined error term dΨ is then estimated. In this regard, there arek, e.g., 2n, values of this error term corresponding to each azimuthcommand with n values on each side of the tracker median. The values forthe combined error term including the estimated tilt parameters are asfollows:

dΨ _(i)=−(VE _(φi)−cos(φ_(ci))sin(γ_(1est))+sin(φ_(ci))sin(γ_(2est)))

The true and measured values of the combined error term dΨ are depictedin FIGS. 16a and 16 b as a function of azimuth and elevation commandangles, respectively.

The estimation of the combined error term dΨ is based on a quadratic fitof the elevation dependant data for dl as defined above. The definitionof intermediate variables to facilitate the solution of the quadraticequations are as follows: $\begin{matrix}{a = {\sum\limits_{i}\psi_{ci}}} & {a = 41.375} & {c = {\sum\limits_{i}\left( \psi_{ci} \right)^{2}}} & {c = 41.557} \\{d = {\sum\limits_{i}\left( \psi_{ci} \right)^{3}}} & {d = 45.749} & {e = {\sum\limits_{i}\left( \psi_{ci} \right)^{4}}} & {e = 53.214} \\{f = {\sum\limits_{i}{d\quad \Psi_{i}}}} & {f = 0.011} & {g = {\sum\limits_{i}{d\quad \Psi_{i}\psi_{ci}}}} & {g = 0.018} \\{h = {\sum\limits_{i}{\left( \psi_{ci} \right)^{2}d\quad \Psi_{i}}}} & {h = 0.025} & \quad & \quad\end{matrix}$

As a result, the quadratic coefficients are:$b_{2} = {\frac{{\left( {{a\quad c} - {kd}} \right)\left( {{af} - {kg}} \right)} - {\left( {a^{2} - {kc}} \right)\left( {{cf} - {kh}} \right)}}{\left( {{a\quad c} - {kd}} \right)^{2} - {\left( {c^{2} - {ke}} \right)\left( {a^{2} - {kc}} \right)}} = 0.00237}$$b_{1} = {\frac{\left( {{cf} - {kh}} \right) - {\left( {c^{2} - {ke}} \right)b_{2}}}{{a\quad c} - {kd}} = {- 0.002599}}$$b_{0} = {\frac{f - {ab}_{1} - {cb}_{2}}{k} = 0.00042}$

Thus, the estimated elevation error is

dΨ _(esti) =b ₀ +b ₁ψ_(ci) +b ₂(ψ_(ci))²

The estimated elevation error can be plotted along with the true andmeasured values of the combined error term dΨ as shown in FIG. 17. Overthe day, the RMS error for dΨ can therefore be defined as follows:${{RM}\quad S_{d\quad \Psi}} = {\sqrt{\frac{1}{k}{\sum\limits_{i}\left( {{d\quad \Psi_{esti}} - {d\quad \Psi_{truei}}} \right)^{2}}} = {0.0647\quad m\quad {rad}}}$

The error terms Φ_(e), δ1 and δ2 are then estimated. In this regard,based upon the left hand side of equation (30), horizontal measurementerrors for negative and positive azimuth commands are defined asfollows:

M _(Φi) =HE_(φi)−sin(ψ_(ci))sin(φ_(ci))sin(γ_(1est))−sin(ψ_(ci))cos(φ_(ci))sin(γ_(2est))

For purposes of illustration, the horizontal measurements errors as afunction of azimuth and elevation command angles are depicted in FIGS.18a and 18 b, respectively.

The substitution values are then calculated as follows: $\begin{matrix}{s = {\sum\limits_{i}M_{\varphi \quad i}}} & {s = {- 0.312}} & {t = {\sum{M_{\varphi \quad i}{\cos \left( \psi_{ci} \right)}}}} & {t = {- 0.201}} \\{u = {\sum\limits_{i}{\cos \left( \psi_{ci} \right)}}} & {u = 29.332} & {v = {\sum\limits_{i}{\cos^{2}\left( \psi_{ci} \right)}}} & {v = 21.040} \\{w = {\sum{M_{\varphi \quad i}{\sin \left( \psi_{ci} \right)}}}} & {w = {- 0.214}} & {x = {\sum\limits_{i}{\sin \left( \psi_{ci} \right)}}} & {x = 34.267} \\{y = {\sum\limits_{i}{{\sin \left( \psi_{ci} \right)}{\cos \left( \psi_{ci} \right)}}}} & {y = {18,272}} & {z = {\sum\limits_{i}{\sin^{2}\left( \psi_{ci} \right)}}} & {z = 26.960}\end{matrix}$

The curve fit coefficients are then calculated using Equations (45)-(47)as follows:$A = {\frac{{\left( {{sx} - {wk}} \right)\left( {{uy} - {vx}} \right)} - {\left( {{tx} - {wu}} \right)\left( {{ky} - {ux}} \right)}}{{\left( {x^{2} - {kz}} \right)\left( {{uy} - {vx}} \right)} - {\left( {{yx} - {uz}} \right)\left( {{ky} - {ux}} \right)}} = {- 0.00214}}$$B = {\frac{{- {A\left( {x^{2} - {kz}} \right)}} + \left( {{sx} - {wk}} \right)}{{ky} - {ux}} = 0.00518}$$C = {\frac{{Az} - {By} - w}{x} = {- {.00181}}}$

The parameters are then estimated: $\begin{matrix}{\delta_{2{est}} = C} & {\delta_{2{est}} = {1.81\quad {mrad}}} & \quad \\{{me}_{\delta 2} = {\delta_{2{est}} - \delta_{2}}} & {{me}_{\delta 2} = {{- 0.19}\quad {mrad}}} & {\frac{{me}_{\delta 2}}{\delta_{2}} = {{- 9.5}\%}} \\{\Phi_{eest} = {a\quad {\sin (B)}}} & {\Phi_{eest} = {5.18\quad {mrad}}} & \quad \\{{me}_{\Phi \quad e} = {\Phi_{eest} - \Phi_{e}}} & {{me}_{\Phi \quad e} = {0.18\quad {mrad}}} & {\frac{{me}\quad \Phi \quad e}{\Phi_{e}} = {3.5\%}} \\{\delta_{1{est}} = {a\quad {\sin (A)}}} & {\delta_{1{est}} = {{- 2.14}\quad {mrad}}} & \quad \\{{me}_{\delta 1} = {\delta_{1{est}} - \delta_{1}}} & {{me}_{\delta 1} = {- 0.14}} & {\frac{{me}_{\delta 1}}{\delta_{1}} = {6.9\%}}\end{matrix}$

Finally, a check of the curve fit to the data can be performed as shownin FIGS. 19a and 19 b and as defined below:

MM _(Φi) =A sin(ψ_(ci))−B cos(ψ_(ci))−C

In the next step, the X, Y and Z estimation errors are determined asfollows:

X _(esti) =T ₁₁(δ_(1est),δ_(2est),

γ_(1est),γ_(2est),λ_(esti),

ω_(esti))cos(ψ_(ci))cos(φ_(ci))+T ₁₂

(δ_(1est),δ_(2est),γ_(1est),γ_(2est),λ_(esti),

ω_(esti))cos(ψ_(ci))sin(φ_(ci))+T ₁₃

(δ_(1est),δ_(2est),γ_(1est),γ_(2est),

λ_(esti),ω_(esti))sin(ψ_(ci))

Y _(esti) =T ₂₁(δ_(1est),δ_(2est),γ_(1est),γ_(2est),λ_(esti),

ω_(esti))cos(ψ_(ci))cos(φ_(ci))+T ₂₂

(δ_(1est),δ_(2est),γ_(1est),γ_(2est),

λ_(esti),ω_(esti))cos(ψ_(ci))sin(φ_(ci))

+T ₂₃(δ_(1est),δ_(2est),γ_(1est),γ_(2est),

λ_(esti),ω_(esti))sin(ψ_(ci))

Z _(esti) =T ₃₁(δ_(1est),δ_(2est),γ_(1est),γ_(2est),λ_(esti),

ω_(esti))cos(ψ_(ci))cos(φ_(ci))+T₃₂(δ_(1est),δ_(2est),γ_(1est),γ_(2est),λ_(esti),

ω_(esti))cos(ψ_(ci))sin(φ_(ci))+T ₃₃

(δ_(1est),δ_(2est),γ_(1est),γ_(2est),λ_(esti),

ω_(esti))sin(ψ_(ci))

wherein

λ_(esti)=ψ_(ci) +dΨ _(esti)

ω_(esti)=φ_(ci)+Φ_(eest)

The estimated vertical measurement RMS error can then be determined asfollows: VE_(ei) = VEm(Z_(esti), X_(esti))${{RM}\quad S_{VE}} = {{\sqrt{\frac{1}{k}{\sum\limits_{i}\left( {{VE}_{ei} - {VE}_{\varphi \quad {ti}}} \right)^{2}}}\quad {RM}\quad S_{VE}} = {0.040\quad {mrad}}}$

The estimated horizontal measurement RMS error can also be determined asfollows: HE_(ei) = HEm(Y_(esti), X_(esti))${{RM}\quad S_{HE}} = {{\sqrt{\frac{1}{k}{\sum\limits_{i}\left( {{HE}_{ei} - {HE}_{\varphi \quad {ti}}} \right)^{2}}}\quad {RM}\quad S_{HE}} = 0.087}$

The vertical and horizontal measurement RMS errors are depicted in FIGS.20a and 20 b, respectively, as a function of azimuth angle. As depicted,the true values of the vertical and horizontal measurement errors andthe estimated values of the vertical and horizontal measurement errorstrack very closely. As this simulation depicts, therefore, the methodand apparatus of the present invention can closely approximate theerrors throughout the day such that the heliostat or other solarconcentrator remains trained on the sun, thereby increasing theefficiency of solar collection.

The final step of the simulation determines the revised open-loopazimuth and elevation commands to reduce the measurement errors usingthe error estimates that were previously determined. This step includestwo parts. The first part verifies that the resulting correctedcommands, using the true errors, results in a near zero measurementerror. The second part uses the estimated errors to compute thecorrected commands and then presents the resulting mathematical errors.

In the first part of this step, the revised open-loop azimuth andelevation commands are computed in order to reduce the measurementerrors using the foregoing error estimates. In this regard, thecorrected elevation rotation is computed using Equation (36) as follows:${\lambda_{cc}\left( {\Psi,\Phi,\gamma_{1},\gamma_{2},{\delta_{1}\delta_{2}}} \right)} = {a\quad {\sin\left\lbrack \frac{\begin{matrix}{{{\sin \left( \gamma_{1} \right)}{\cos (\Psi)}{\cos (\Phi)}{\cos \left( \gamma_{2} \right)}} - {{\sin \left( \gamma_{2} \right)}{\cos (\Psi)}{\sin (\Phi)}} +} \\{{{\cos \left( \gamma_{1} \right)}{\cos \left( \gamma_{2} \right)}{\sin (\Psi)}} - {{\sin \left( \delta_{1} \right)}{\sin \left( \delta_{2} \right)}}}\end{matrix}}{{\cos \left( \delta_{1} \right)}{\cos \left( \delta_{2} \right)}} \right\rbrack}}$

while the corrected azimuth rotation is calculated from Equation (21) asfollows:${\cos \quad {\omega \left( {A,B,C,D} \right)}} = \frac{{A\quad C} + {BD}}{A^{2} + B^{2}}$

The corrected elevation command history using the true errors is thendetermined as follows:

λ_(cti)=λ_(cc)(ψ_(ci),φ_(ci),γ₁,γ₂,δ₁,δ₂)

The corrected azimuth command history using the true errors is then alsodetermined as follows:

A _(i)=cos(λ_(cti))cos(δ₂)

B _(i)=cos(δ₁)sin(δ₂)−sin(δ₁)cos(δ₂)sin(λ_(cti))

C _(i)=cos(γ₁)cos(ψ_(ci))cos(φ_(ci))−sin(ψ_(ci))sin(γ₁)

D_(i)=sin(γ₁)sin(γ₂)cos(ψ_(ci))cos(φ_(ci))+cos(γ₂)cos(ψ_(ci))sin(φ_(ci))+sin(ψ_(ci))cos(γ₁

ω_(cti) =a cos[cos ω(A _(i) ,B _(i) ,C _(i) ,D _(i))]

The vertical and horizontal errors are then determined using the revisedcommands with no noise as follows:

X _(cori) =T ₁₁(δ₁,δ₂,γ₁,γ₂,λ_(cti),

ω_(cti))cos(ψ_(ci))cos(φ_(ci))+T ₁₂

(δ₁,δ₂,γ₁,γ₂,λ_(cti),

ω_(cti))cos(ψ_(ci))sin(φ_(ci))+T ₁₃

(δ₁,δ₂,γ₁,γ₂,λ_(cti),

ω_(cti))sin(ψ_(ci))

Y _(cori) =T ₂₁(δ₁,δ₂,γ₁,γ₂,λ_(cti),

ω_(cti))cos(ψ_(ci))+T ₂₂

(δ₁,δ₂,γ₁,γ₂,

λ_(cti),ω_(cti))cos(ψ_(ci))sin(φ_(ci))

+T ₂₃(δ₁,δ₂,γ₁,γ₂,

λ_(cti),ω_(cti))sin(ψ_(ci))

Z _(cori) =T ₃₁(δ₁,δ₂,γ₁,γ₂,λ_(cti),

ω_(cti))cos(ψ_(ci))cos( φ_(ci))+T ₃₂

(δ₁,δ₂,γ₁,γ₂,λ_(cti),

ω_(cti))cos(ψ_(ci))sin(φ_(ci))+T ₃₃

(δ₁,δ₂,γ₁,γ₂,

λ_(cti),ω_(cti))sin(ψ_(ci))

The resulting vertical error is depicted in FIG. 21 a and is defined asfollows:

VED _(Φcorri) =VEm(Z _(cori) ,X _(cori))

Likewise, the horizontal error is depicted in FIG. 21b and is defined asfollows:

HE _(Φcorri) =HEm(Y_(cori) ,X _(cori))

In the second part of this step, the corrected azimuth and elevationcommands are determined using the estimated errors and the resultingvertical and horizontal measurement errors are then assessed. In thisinstance, it should be noted that only residual errors are utilized inthe equations for the vertical and horizontal error estimates.

Initially, the corrected elevation command history is determined usingthe estimated errors as follows:

λ_(cei)=λ_(cc)(ψ_(ci),φ_(ci),γ_(1est),γ_(2est),δ_(1est),δ_(2est))

The corrected azimuth command history is also determined using theestimated errors as follows:

A _(i)=cos(λ_(cei))cos(δ_(2est))

B _(i)=cos(δ_(1est))sin(δ_(2est))−sin(δ_(1est))cos(δ_(2est))sin(λ_(cei))

C _(i)=cos(γ_(1est))cos(ψ_(ci))cos(φ_(ci))−sin(ψ_(ci))sin(γ_(1est))

D_(i)=sin(γ_(1est))sin(γ_(2est))cos(ψ_(ci))cos(φ_(ci))+cos(γ_(2est))cos(ψ_(ci))sin(φ_(ci))+sin(ψ_(ci))cos(γ_(1est))sin(γ_(2est))

ω_(cei) =a cos[cos ω(A _(i) ,B _(i) ,C _(i) ,D _(i))]

The vertical and horizontal errors are then determined using the revisedcommands with no noise as follows:

X _(cor1i) =T ₁₁(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))cos(φ_(ci))+T ₁₂

(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))sin(φ_(ci))+T ₁₃

(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))sin(ψ_(ci))

Y _(cor1i) =T ₂₁(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))cos(φ_(ci))+T ₂₂

(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))sin(φ_(ci))+T ₂₃

(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))sin(ψ_(ci))

Z _(cor1i) =T ₃₁(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))cos(φ_(ci))+T ₃₂

(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))sin(φ_(ci))+T ₃₃

(δ₁,δ₂,γ₁,γ₂,λ_(cei)

ω_(cei))cos(ψ_(ci))cos(φ_(ci))sin(ψ_(ci))

As shown in FIG. 22a, the vertical error is defined as:

VE _(Φcorri) =VEm(Z _(cor1i) ,X _(cor1i))

As shown in FIG. 22b, the horizontal error is also defined as:

HE _(Φcorri) =HEm(Y _(cor1i),X_(cor1i))

Finally, the RMS Error can be determined using uncorrected commands asfollows: $\begin{matrix}{{Vertical}\quad {error}} & {{RMS2}_{VE} = \sqrt{\frac{1}{k}{\sum\limits_{i}\left( {VE}_{\varphi \quad i} \right)^{2}}}} & {{RMS2}_{VE} = {4.77\quad {mrad}}} \\{{Horizontal}\quad {error}} & {{RMS2}_{HE} = \sqrt{\frac{1}{k}{\sum\limits_{i}\left( {HE}_{\varphi \quad i} \right)^{2}}}} & {{RMS2}_{HE} = {6.793\quad {mrad}}}\end{matrix}$

In addition, the RMS error can be determined using corrected commands asfollows: $\begin{matrix}{{Vertical}\quad {error}} & {{RMS1}_{VE} = \sqrt{\frac{1}{k}{\sum\limits_{i}\left( {VE}_{\varphi \quad {corri}} \right)^{2}}}} & {{RMS1}_{VE} = {0.055\quad {mrad}}} \\{{Horizontal}\quad {error}} & {{RMS1}_{HE} = \sqrt{\frac{1}{k}{\sum\limits_{i}\left( {HE}_{\varphi \quad {corri}} \right)^{2}}}} & {{RMS1}_{HE} = {0.128\quad {mrad}}}\end{matrix}$

As shown, refining the azimuth and elevation commands based upon theerrors from a prior day significantly improves the performance of thesolar concentrator on subsequent days since the solar concentrator willremain trained more closely upon the sun. As such, the method andapparatus of the present invention permits a solar concentrator tocollect a greater percentage of the incident light.

Many modifications and other embodiments of the invention will come tomind to one skilled in the art to which this invention pertains havingthe benefit of the teachings presented in the foregoing descriptions andthe associated drawings. Therefore, it is to be understood that theinvention is not to be limited to the specific embodiments disclosed andthat modifications and other embodiments are intended to be includedwithin the scope of the appended claims. Although specific terms areemployed herein, they are used in a generic and descriptive sense onlyand not for purposes of limitation.

What is claimed is:
 1. A method of controllably positioning a solarconcentrator comprising: determining a vertical error and a horizontalerror between a centerline of the solar concentrator and a sun referencevector; determining an elevation command and an azimuth command tocompensate for the vertical error and the horizontal error, whereindetermining the elevation command and the azimuth command comprises:determining respective errors generated by a plurality of error sourcesthat contribute to at least one of the vertical error and the elevationerror, wherein the plurality of error sources include at least one errorsource selected from a system elevation error group consisting of agravitational residue error g, an elevation transfer function error eand an error r due to atmospheric refraction, and wherein determiningthe respective errors comprises determining a respective contribution toat least one of the vertical error and the horizontal error that isattributable to at least one error source selected from the systemelevation error group and that is independent of any contribution fromany error source outside of the system elevation error group; anddetermining the elevation command and the azimuth command at leastpartially based upon the respective errors generated by the plurality oferror sources; and positioning the solar concentrator based upon theelevation command and the azimuth command in order to compensate for thevertical error and the horizontal error.
 2. A method according to claim1 wherein said determining and positioning steps are repeated at aplurality of different times throughout a day.
 3. A method according toclaim 1 wherein determining the vertical error and the horizontal errorcomprises: moving the solar concentrator from a nominal position to analigned position at which a difference between gas temperatures of eachquadrant of the solar concentrator is minimized; and determining thehorizontal and vertical distances that the solar concentrator is movedfrom the nominal position to the aligned position.
 4. A method accordingto claim 1 wherein determining the vertical error and the horizontalerror comprises: moving the solar concentrator from a nominal positionto an aligned position at which a maximum power factor is obtained; anddetermining the horizontal and vertical distances that the solarconcentrator is moved from the nominal position to the aligned position.5. A method according to claim 1 wherein determining the elevationcommand comprises determining an elevation command angle Ψ_(c) asfollows: Ψ_(c)=sin⁻¹(sin γ₁ cos Ψ_(r) cos Φ_(r)−sin γ₂ cos Ψ_(r) sinΦ_(r)+sin Ψ_(r))−dΨ wherein γ₁ and γ₂ are azimuth rotational tilterrors, Φ_(r) and Ψ_(r) are elevation and azimuth angles, respectively,in an inertial reference system that are required to align thecenterline of the solar concentrator and a sun reference vector, and dΨis a combination of the elevation transfer function error, thegravitational residual error, and the error due to atmosphericrefraction.
 6. A method according to claim 1 wherein determining theazimuth command comprises determining an azimuth command angle Φ_(c) asfollows:$\Phi_{c} = {{\cos^{- 1}\left\lbrack \frac{{A\quad C} + {BD}}{A^{2} + B^{2}} \right\rbrack} - \Phi_{e}}$

wherein A, B, C and D are defined as follows: A=cos λ cos δ₂ B=cos δ₁sin δ₂−sin δ₁ cos δ₂ sinλ C=cos γ₁ cos Ψ₁ cos Φ_(r)−sin γ₁ sin Ψ_(r)D=sin γ₁ sin γ₂ cos Ψ_(r) cos Φ_(r)+cos γ₂ cos Ψ_(r) sin Φ_(r)+sin Ψ_(r)cos γ₁ sin γ₂ wherein λ is defined as follows: λ=Ψ_(c)−Ψ_(e)+g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c)) wherein Ψ_(c) is an elevation command angle,Ψ_(e) is an elevation reference position error, gΨ_(c) is agravitational residue error, eΨ_(c) is an elevation transfer functionerror and rΨ_(c) is an error due to atmospheric refraction, and whereinδ₁ is an elevation rotational tilt error, δ₂ is a reflective surfacenon-orthogonality error, γ₁ and γ₂ are azimuth rotational tilt errorsand Ψ_(r) and Φ_(r) are elevation and azimuth angles, respectively, inan inertial reference system that are required to align the centerlineof the solar concentrator and a sun reference vector.
 7. A methodaccording to claim 1 wherein determining the vertical error and thehorizontal error comprises obtaining sun pointing error measurementinformation with at least one of a sun sensor and a digital imageradiometer.
 8. A method of controllably positioning a solar concentratorcomprising: determining a vertical error and a horizontal error betweena centerline of the solar concentrator and a sun reference vector;determining an elevation command and an azimuth command to compensatefor the vertical error and the horizontal error, wherein determining theelevation command and the azimuth command comprises: determiningrespective errors generated by a plurality of error sources thatcontribute to at least one of the vertical error and the elevationerror, wherein the plurality of error sources include at least one errorsource selected from the group consisting of a gravitational residueerror g, an elevation transfer function e and an error r due toatmospheric refraction, and wherein determining the respective errorscomprises individually determining each of a first azimuth rotationaltilt error γ₁, a second azimuth rotational tilt error γ₂, a firstelevation rotational tilt error Φ_(e), a second elevation rotationaltilt error δ₁, and a reflective surface non-orthogonality error δ₂; anddetermining the elevation command and the azimuth command at leastpartially based upon the respective errors generated by the plurality oferror sources; and positioning the solar concentrator based upon theelevation command and the azimuth command in order to compensate for thevertical error and the horizontal error.
 9. A method of controllablypositioning a solar concentrator comprising: determining a verticalerror and a horizontal error between a centerline of the solarconcentrator and a sun reference vector; determining an elevationcommand and an azimuth command to compensate for the vertical error andthe horizontal error, wherein determining the elevation command and theazimuth command comprises: determining respective errors generated by aplurality of error sources that contribute to at least one of thevertical error and the elevation error, wherein the plurality of errorsources include at least one error source selected from the groupconsisting of a gravitational residue error g, an elevation transferfunction error e and an error r due to atmospheric refraction, andwherein determining the respective errors comprises collectivelydetermining the gravitational residue error g, the elevation transferfunction error e and the error r due to atmospheric refraction; anddetermining the elevation command and the azimuth command at leastpartially based upon the respective errors generated by the plurality oferror sources; and positioning the solar concentrator based upon theelevation command and the azimuth command in order to compensate for thevertical error and the horizontal error.
 10. An apparatus forcontrollably positioning a solar concentrator comprising: a measurementsystem for determining a vertical error and a horizontal error between acenterline of the solar concentrator and a sun reference vector; aprocessing element, responsive to said measurement system, fordetermining an elevation command and an azimuth command to compensatefor the vertical error and the horizontal error, wherein said processingelement determines respective errors generated by a plurality of errorsources that contribute to at least one of the vertical error and theelevation error, wherein the plurality of error sources include at leastone error source selected from a system elevation error group consistingof a gravitational residue error g, an elevation transfer function errore and an error r due to atmospheric refraction, wherein said processordetermines the respective errors by determining a respectivecontribution to at least one of the vertical error and the horizontalerror that is attributable to at least one error source selected fromthe system elevation error group and that is independent of anycontribution from any error source outside of the system elevation errorgroup, and wherein said processing element determines the elevationcommand and the azimuth command at least partially based upon therespective errors generated by the plurality of error sources; and apositioning mechanism, responsive to said processing element, forpositioning the solar concentrator based upon the elevation command andthe azimuth command in order to compensate for the vertical error andthe horizontal error.
 11. An apparatus according to claim 10 whereinsaid measurement system repeatedly determines the vertical error and thehorizontal error at a plurality of different times throughout a day. 12.An apparatus according to claim 10 wherein said processing elementdetermines the respective errors by individually determining each of afirst azimuth rotational tilt error γ₁, a second azimuth rotational tilterror γ₂, a first elevation rotational tilt error Φ_(e), a secondelevation rotational tilt error δ₁, and a reflective surfacenon-orthogonality error δ₂.
 13. An apparatus according to claim 10wherein said measurement system is adapted to move the solarconcentrator from a nominal position to an aligned position at which amaximum power factor is obtained, and wherein said measurement system isfurther adapted to determine the horizontal and vertical distances thatthe solar concentrator is moved from the nominal position to the alignedposition.
 14. An apparatus according to claim 10 wherein said processingelement determines the elevation command by determining an elevationcommand angle Φ_(c) as follows: Ψ_(c)=sin⁻¹(sin γ₁ cos Ψ_(r) cosΦ_(r)−sin γ₂ cos Ψ_(r) sin Φ_(r)+sin Ψ_(r))−dΨ wherein γ₁ and γ₂ areazimuth rotational tilt errors, Ψ_(r) and Φ_(r) are elevation andazimuth angles, respectively, in an inertial reference system that arerequired to align the centerline of the solar concentrator and a sunreference vector, and dΨ is a combination of the elevation transferfunction error, the gravitational residual error, and the error due toatmospheric refraction.
 15. An apparatus according to claim 10 whereinsaid processing element determines the azimuth command by determining anazimuth command angle Φ_(r), as follows:$\Phi_{c} = {{\cos^{- 1}\left\lbrack \frac{{A\quad C} + {BD}}{A^{2} + B^{2}} \right\rbrack} - \Phi_{e}}$

wherein A, B, C and D are defined as follows: A=cos λ cos δ₂ B=cos δ₁sin δ₂−sin δ₁ cos δ₂ sin λ C=cos γ₁ cos Ψ₁ cos Φ_(r)−sin γ₁ sin Ψ_(r)D=sin γ₁ sin γ₂cos Ψ_(r) cos Φ_(r)+cos γ₂ cos Ψ_(r) sin Φ_(r)+sin Ψ_(r)cos γ₁ sin γ₂ wherein λ is defined as follows: λ=Ψ_(c)−Ψ_(e)+g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c)) wherein Ψ_(c) is an elevation command angle,Ψ_(e) is an elevation reference position error, gΨ_(c) is agravitational residue error, eΨ_(c) is an elevation transfer functionerror and rΨ_(c) is an error due to atmospheric refraction, and whereinδ₁ is an elevation rotational tilt error, δ₂ is a reflective surfacenon-orthogonality error, γ₁ and γ₂ are azimuth rotational tilt errorsand Ψ_(r) and Φ_(r) are elevation and azimuth angles, respectively, inan inertial reference system that are required to align the centerlineof the solar concentrator and a sun reference vector.
 16. An apparatusaccording to claim 10 wherein said measurement system is selected fromthe group consisting of a sun sensor and a digital image radiometer. 17.An apparatus according to claim 10 wherein said measurement system isadapted to move the solar concentrator from a nominal position to analigned position at which a difference between gas temperatures of eachquadrant of the solar concentrator is minimized and wherein saidmeasurement system is further adapted to determine the horizontal andvertical distances that the solar concentrator is moved from the nominalposition to the aligned position.
 18. An apparatus for controllablypositioning a solar concentrator comprising: a measurement system fordetermining a vertical error and a horizontal error between a centerlineof the solar concentrator and a sun reference vector; a processingelement, responsive to said measurement system, for determining andelevation command and an azimuth command to compensate for the verticalerror and the horizontal error, wherein said processing elementdetermines respective errors generated by a plurality of error sourcesthat contribute to at least one of the vertical error source selectedfrom the group consisting of a gravitational residue error g, anelevation transfer function error e and an error r due to atmosphericrefraction, wherein said processing element determines the respectiveerrors by collectively determining the gravitational residue error g,the elevation transfer function error e and the error r due toatmospheric refraction, and wherein said processing element determinesthe elevation command and the azimuth command at least partially basedupon the respective errors generated by the plurality of error sources;and a positioning mechanism, responsive to said processing element, forpositioning the solar concentrator based upon the elevation commend andthe azimuth command in order to compensate for the vertical error andthe horizontal error.
 19. A control system for positioning a solarconcentrator comprising: an input section for receiving signalsrepresentative of a vertical error and a horizontal error between acenterline of the solar concentrator and a sun reference vector; aprocessing element, responsive to said input section, for determining anelevation command and an azimuth command to compensate for the verticalerror and the horizontal error, wherein said processing elementdetermines respective errors generated by a plurality of error sourcesthat contribute to at least one of the vertical error and the elevationerror, wherein the plurality of error sources include at least one errorsource selected from a system elevation error group consisting of agravitational residue error g, an elevation transfer function error eand an error r due to atmospheric refraction, and wherein saidprocessing element determines the respective errors by determining arespective contribution to at least one of the vertical error and thehorizontal error that is attributable to at least one error sourceselected from the system elevation error group and that is independentof ant contribution from any error source outside of the systemelevation error group, and wherein said processing element determinesthe elevation command and the azimuth command at least partially basedupon the respective errors generated by the plurality of error sources;and an output section, responsive to said processing element, forproviding signals representative of the elevation command and theazimuth command in order to controllably position the solar concentratorso as to compensate for the vertical error and the horizontal error. 20.A control system according to claim 19 wherein said input sectionrepeatedly receives signals representative of the vertical error and thehorizontal error at a plurality of different times throughout a day. 21.A control system according to claim 19 wherein said processing elementdetermines the elevation command by determining an elevation commandangle Ψc as follows: Ψ_(c)=sin⁻¹(sin γ₁ cos Ψ_(r) cos Φ_(r)−sin γ₂ cosΨ_(r) sin Φ_(r)+sin Ψ_(r))−dΨ wherein γ₁ and γ₂ are azimuth rotationaltilt errors, Ψ_(r) and Φ_(r) are elevation and azimuth angles,respectively, in an inertial reference system that are required to alignthe centerline of the solar concentrator and a sun reference vector, anddΨ is a combination of the elevation transfer function error, thegravitational residual error, and the error due to atmosphericrefraction.
 22. A control system according to claim 19 wherein saidprocessing element determines the azimuth command by determining anazimuth command angle Φ_(c) as follows:$\Phi_{c} = {{\cos^{- 1}\left\lbrack \frac{{A\quad C} + {BD}}{A^{2} + B^{2}} \right\rbrack} - \Phi_{e}}$

wherein A, B, C and D are defined as follows: A=cos λ cos δ₂ B=cos δ₁sin δ₂−sin δ₁ cos δ₂ sin λ C=cos γ₁ cos Ψ₁ cos Φ_(r)−sin γ₁ sin Ψ_(r)D=sin γ₁ sin γ₂ cos Ψ_(r) cos Φ_(r)+cos γ₂ cos Ψ_(r) sin Φ_(r)+sin Ψ_(r)cos γ₁ sin γ₂ wherein λ is defined as follows: λ=Ψ_(c)Ψ_(e)+g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c)) wherein Ψ_(c) is an elevation command angle,Ψ_(e) is an elevation reference position error, gΨ_(c) is agravitational residue error, eΨ_(c) is an elevation transfer functionerror and rΨ_(c) is an error due to atmospheric refraction, and whereinδ₁ is an elevation rotational tilt error, δ₂ is a reflective surfacenon-orthogonality error, γ₁ and γ₂ are azimuth rotational tilt errorsand Ψ_(r) and Φ_(r) are elevation and azimuth angles, respectively, inan inertial reference system that are required to align the centerlineof the solar concentrator and a sun reference vector.
 23. A controlsystem for positioning a solar concentrator comprising: an input sectionfor receiving signals representative of a vertical error and ahorizontal error between a centerline of the solar concentrator and asun reference vector; a processing element, responsive to said inputsection, for determining an elevation command and an azimuth command tocompensate for the vertical error and the horizontal error, wherein saidprocessing element determines respective errors generated by a pluralityof error sources that contribute to at least one of the vertical errorand the elevation error, wherein the plurality of error sources includeat least one error source selected from the group consisting of agravitational residue error g, and elevation transfer function error eand an error r due to atmospheric refraction, wherein said processingelement determines the respective errors by individually determiningeach of a first azimuth rotational tilt error γ₁, a second azimuthrotational tilt error γ₂, a first elevation rotational tilt error Φ_(e),a second elevation rotational tilt error δ₁, and a reflective surfacenon-orthogonality error δ₂, and wherein said processing elementdetermines the elevation command and the azimuth command and the azimuthcommand at least partially based upon the respective errors generated bythe plurality of error sources; and an output section, responsive tosaid processing element, for providing signals representative of theelevation command and the azimuth command in order controllably positionthe solar concentrator so as to compensate for the vertical error andthe horizontal error.
 24. A control system for positioning a solarconcentrator comprising: an input section for receiving signalsrepresentative of a vertical error and a horizontal error between acenterline of the solar concentrator and a sun reference vector; aprocessing element, responsive to said input section, for determining anelevation command and an azimuth command to compensate for the verticalerror and the horizontal error, wherein said processing elementdetermines respective errors generated by a plurality of error sourcesthat contribute to at least one of the vertical error and the elevationerror, wherein the plurality of error sources include at least one errorsource selected from the group consisting of a gravitational residueerror g, an elevation transfer function error e and an error r due toatmospheric refraction, wherein said processing element determines therespective errors by collectively determining the gravitational residueerror g, the elevation transfer function error e and the error r due toatmospheric refraction, and wherein said processing element determinesthe elevation command and the azimuth command at least partially basedupon the respective errors generated by the plurality of error sources;and an output section, responsive to said processing element, forproviding signals representative of the elevation command and theazimuth command in order to controllably position the solar concentratorso as to compensate for the vertical error and the horizontal error. 25.A computer program product for positioning a solar concentrator, thecomputer program product comprising a computer-readable storage mediumhaving computer-readable program code embodied in said medium, saidcomputer-readable program code comprising: a first computer-readableprogram code portion for determining an elevation command and an azimuthcommand to compensate for the vertical error and the horizontal errorbetween a centerline of the solar concentrator and a sun referencevector, wherein said first computer-readable program code portioncomprises: a second computer-readable program code portion fordetermining respective errors generated by a plurality of error sourcesthat contribute to at least one of the vertical error and the elevationerror, wherein the plurality of error sources include at least one errorsource selected from a system elevation error group consisting of thegravitational residue error g, the elevation transfer function error eand the error r due to atmospheric refraction, and wherein said secondcomputer-readable program code portion determines the respective errorsby determining a respective contribution to at least one of the verticalerror and the horizontal error that is attributable to at least oneerror source selected from the selected from the system elevation errorgroup and that is independent of any contribution from any error sourceoutside of the system elevation error group; and a thirdcomputer-readable program code portion, responsive to said secondcomputer-readable program code portion for determining the elevationcommand and the azimuth command at least partially based upon therespective errors generated by the plurality of error sources, whereinthe elevation command and the azimuth command are capable ofcontrollably positioning the solar concentrator so as to compensate forthe vertical error and the horizontal error.
 26. A computer programproduct according to claim 25 wherein said first computer-readableprogram code portion repeatedly determines the elevation command and theazimuth command in response to different vertical errors and horizontalerrors at a plurality of different times throughout a day.
 27. Acomputer program product according to claim 25 wherein said secondcomputer-readable program code portion determines the respective errorsby individually determining each of a first azimuth rotational tilterror γ₁, a second azimuth rotational tilt error γ₂, a first elevationrotational tilt error Φ_(e), a second elevation rotational tilt errorδ₁, and a reflective surface non-orthogonality error δ_(2.)
 28. Acomputer program product according to claim 25 wherein said thirdcomputer-readable program code portion determines the azimuth command bydetermining an azimuth command angle Φ_(c) as follows:$\Phi_{c} = {{\cos^{- 1}\left\lbrack \frac{{A\quad C} + {BD}}{A^{2} + B^{2}} \right\rbrack} - \Phi_{e}}$

wherein A, B, C and D are defined as follows: A=cos λ cos δ₂ B=cos δ₁sin δ₂−sin δ₁ cos δ₂ sin λ C=cos γ₁ cos Ψ₁ cos Φ_(r)−sin γ₁ sin Ψ_(r)D=sin γ₁ sin γ₂ cos Ψ_(r) cos Φ_(r)+cos γ₂ cos Ψ_(r) sin Φ_(r)+sin Ψ_(r)cos γ₁ sin γ₂ wherein λ is defined as follows: λ=Ψ_(c)−Ψ_(e)+g(Ψ_(c))+e(Ψ_(c))+r(Ψ_(c)) wherein Ψ_(c) is an elevation command angle,Ψ_(e) is an elevation reference position error, gΨ_(c) is agravitational residue error, eΨ_(c) is an elevation transfer functionerror and rΨ_(c) is an error due to atmospheric refraction, and whereinδ₁ is an elevation rotational tilt error, δ₂ is a reflective surfacenon-orthogonality error, γ₁ and γ₂ are azimuth rotational tilt errorsand Ψ_(r) and Φ_(r) are elevation and azimuth angles, respectively, inan inertial reference system that are required to align the centerlineof the solar concentrator and a sun reference vector.
 29. A computerprogram product according to claim 25 wherein said thirdcomputer-readable program code portion determines the elevation commandby determining an elevation command angle Ψ_(c) as follows:Ψ_(c)=sin⁻¹(sin γ₁ cos Ψ_(r) cos Φ_(r)−sin γ₂ cos Ψ_(r) sin Φ_(r)+sinΨ_(r))−dΨ wherein γ₁ and γ₂ are azimuth rotational tilt errors, Ψ_(r)and Φ_(r) are elevation and azimuth angles, respectively, in an inertialreference system that are required to align the centerline of the solarconcentrator and a sun reference vector, and dΨ is a combination of theelevation transfer function error, the gravitational residual error, andthe error due to atmospheric refraction.
 30. A computer program productfor positioning a solar concentrator, the computer comprising acomputer-readable storage medium having computer-readable programproduct comprising a computer-readable storage medium havingcomputer-readable program code embodied in said medium, saidcomputer-readable program code comprising: a first computer-readableprogram code portion for determining an elevation command and an azimuthcommand to compensate for the vertical error and the horizontal errorbetween a centerline of the solar concentrator and a sun referencevector, wherein said first computer-readable program code portioncomprises: a second computer-readable program code portion fordetermining respective errors generated by a plurality of error sourcesthat contribute to at least one of the vertical error and the elevationerror, wherein the plurality of error sources include at least one errorsource selected from the group consisting of the gravitational residueerror g, the elevation transfer function error e and the error r due toatmospheric refraction, wherein said second computer-readable programcode portion determines the respective errors by collectivelydetermining the gravitational residue error g, the elevation transferfunction error e and the error r due to atmospheric refraction; and athird computer-readable program code portion, responsive to said secondcomputer-readable program code portion for determining the elevationcommand and the azimuth command at least partially based upon therespective errors generated by the plurality of error sources, whereinthe elevation command and the azimuth command are capable ofcontrollably positioning the solar concentrator so as to compensate forthe vertical error and the horizontal error.